UNIVERSITY  OFCALIFORNSA 
AT    LOS  ANGELES 


The  volumes  of  the  University  of  Michigan 
Studies  are  published  by  authority  of  the  Executive 
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thus  far  published  is  given  at  the  end  of  this  volume. 


Winixftxsitvi  of  pCicMgan  ^tWidijcs 

SCIENTIFIC  SERIES 

VOLUME    II 


STUDIES  ON  DIVERGENT  SERIES  AND  SUMMABILITY 


^T^^ 


THE  MACMILLAN   COMPANY 

NEW  YORK  BOSTON  CHICAGO 

DALLAS  SAN    FRANCISCO 

MACMILLAN  &  CO..  Limited 

LONDON      BOMBAY       CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA.  Ltd. 

TORONTO 


STUDIES  ON 
DIVERGENT  SERIES  AND  SUMMABILITY 


BY 

WALTER  BURTON  FORD,  Ph.D. 


MICHIGAN  SCIENCE  SERIES-VOL.  II 


Btto  Pork 

THE  MACMILLAN  COMPANY 

1916 

All  rights  reserved 


Copyright,  1916 
By  The  University  of  Michigan 


PRESS  OF 

THE  NEW  ERA  PRfNTING  COMPANY 

LANCAiTES,  PA. 


LJOrary 


To  My  Fathek 

SYLVESTER  FORD 

This  Book  is  Gratefully  Dedicated. 


Ill 


CONTENTS 

Page 

Chapter  I.     The    Maclaurin    Sum-Formula,    with    Introduction    to    the 

Study  of  Asymptotic  Series 1 

Chapter  IL     The  Determination  of  the  Asymptotic  Developments  of  a 

Given  Function -, '^-■• 

Chapter  III.     The  Asymptotic  Solutions  of  Linear  Differential  Equations.     64 

Chapter  IV.     Elementary  Studies  on  the  Summability  of  Series 75 

Chapter  V.     The  Summability  and  Convergence  of  Fourier  Series  and 

Allied  Developments 102 

Appendix 1' ^ 

Bibliography 184 


PREFACE 

During  the  academic  year  1908-9  the  author  was  privileged  to  give  as  a 
part  of  his  work  at  the  University  of  Michigan  a  course  of  lectures  on  infinite 
series,  with  especial  reference  in  the  second  semester  to  divergent  series  —  a 
subject  which,  despite  the  uncertain  value  so  long  attached  to  it,  seemed  clearly 
to  be  coming  into  increasing  prominence  and  importance  in  mathematical 
analysis.  Little  was  accomplished,  however,  as  regards  divergent  series  beyond 
the  merest  beginning;  yet  this  was  sufiicient  to  awaken  a  desire  to  continue 
farther  and  this  in  turn  resulted  in  a  course  being  given  throughout  the  whole 
of  the  following  year  devoted  entirely  to  divergent  series  and  the  related  topic 
of  summability.  But  this  year  also  closed  with  much  less  ground  satisfactorily 
covered  than  had  been  expected,  unforeseen  difficulties  having  arisen  from  time 
to  time,  some  due  to  the  inherent  complexities  of  the  subject  in  hand  and  others 
to  the  somewhat  hastily  conceived  and  hence  unsatisfactory  state  in  which 
much  of  the  related  literature  was  found  to  be.  Thus  the  course  still  seemed 
altogether  incomplete.  It  was  therefore  decided  to  continue  it  once  more 
throughout  the  following  year,  1910-11,  and  indeed  for  a  like  reason  it  was 
finally  continued  throughout  1911-12.  As  the  lectures  and  class-room  dis- 
cussions progressed,  permanent  notes  were  kept  in  the  hope  that  the  whole  might 
possibly  pass  through  the  press  at  some  future  time  and  appear  in  book  form — 
a  hope  which,  after  various  delays  during  which  the  original  notes  have  been 
considerably  supplemented,  now  reaches  its  realization  in  the  appearance  of  the 
present  volume.  In  its  final  form  it  certainly  presents  a  large  mass  of  detail 
and  is  doubtless  open  to  criticism  in  many  respects,  but  it  does  not  seem  advisable 
to  attempt  any  further  defence  for  it  than  is  contained  in  the  remaining  sections 
of  this  preface  wherein,  after  certain  generalities,  the  content  and  motive  of  the 
various  chapters  are  discussed  in  some  detail. 

Speaking  roughly,  the  study  of  divergent  series,  at  least  as  the  author  has 
come  to  conceive  of  it,  may  be  divided  into  two  parts,  the  one  concerning  the- 
so-called  asymptotic  series  and  the  other  the  theory  of  summability.  Of  these  the 
first,  representing  the  older  aspect,  originated  in  an  isolated  note  by  Cauchy 
in  1843^  relating  to  the  well-known  series  of  Stirling  for  log  T(x),  viz.: 

(1)     \ognx)  =  ilog2r+(x-i}logx-x+^'^l-  §^^^,  +  ^l^,-  .... 

{Bm  =  mih  Bernoulli  number.) 

Cauchy  pointed  out  that  this  series,  though  divergent  for  all  values  of  x,  may  be 

1  "Sur  I'emploi  legitime  des  series  divergentes,"  Coinpt.  Rend,  de  I'Acad.  des  Sciences,  Vol.  17, 
pp.  370-376. 

vii 


viii  Preface 

used  in  computing  log  r(.r)  when  .-r  is  large  (and  positive) — in  fact,  it  was  shown 
that,  having  fixed  the  number  n  of  terms  taken,  the  absolute  error  committed  by 
stopping  the  summation  at  the  7ith  term  is  less  than  the  absolute  value  of  the 
next  succeeding  term,  and  hence  becomes  arbitrarily  small  (n  >  3)  with  in- 
creasing X.  Cauchy's  work  on  divergent  series  was  confined,  however,  to  the 
single  series  (1)  and,  owing  to  the  emphasis  placed  upon  convergent  processes 
exclusivel}'  by  the  successors  of  Cauchy  and  Abel,  no  further  progress  was  made 
in  this  interesting  field  until  the  subject  at  last  reappeared  after  more  than  forty 
years  in  connection  with  the  researches  of  Poincare  upon  the  irregular  solutions 
of  linear  differential  equations.-  Poincare  considered  those  divergent  series 
(normal  series)  of  the  form 

„  ,     ,  /     .        ,  „   .  s  fix)  =  volynomial  in  x, 

which  for  some  time  had  been  known  to  satisfy  formally  linear  differential  equa- 
tions of  certain  types  having  the  point  a:  =  oo  as  an  "  irregular  "  point,  and  he 
showed  essentially  that  in  general  to  every  such  formal  solution  there  corre- 
sponds an  actual  solution  which  can  be  represented  by  (2)  in  much  the  same  sense 
as  (1)  was  described  above  as  representing  log  T{x)}  In  view  of  the  important 
significance  of  such  results  both  from  the  standpoint  of  the  possible  use  of  di- 
vergent series  as  well  as  from  that  of  the  theory  of  differential  equations,  Poin- 
care set  apart  and  discussed  in  some  detail  a  broad  class  of  divergent  series  of 
the  special  form  (2),  applying  to  them  the  name  of  "  asymptotic  series."  PoiN- 
care's  results,  however,  in  so  far  as  they  concerned  differential  equations,  were 
noticeably  incomplete,  being  limited  by  certain  unfortunate  restrictions,  and  thus 
his  original  studies  have  given  rise  in  later  years  to  numerous  researches,  notably 
by  Horn,  in  which  noteworthy  advances  have  been  made,  though  open  questions 
in  this  connection  still  remain.  Corresponding  investigations  (likewise  begun  by 
Poincare)  pertaining  to  linear  difference  equations  have  been  undertaken  in 
recent  years  and  carried  to  an  advanced  stage  by  Horn,  Norlund,  and  others. 
Meanwhile  an  important  aspect  of  the  theory  of  asymptotic  series  has  come  into 
\'iew,  especially  in  England  under  the  leadership  of  Barnes  and  Hardy;  namely, 
that  of  actually  determining  the  asymptotic  developments  of  a  given  function — 
a  problem  of  decided  interest  for  the  study  and  classification  of  functions  in  gen- 
eral. This  latter  aspect  of  the  subject  presents  a  high  degree  of  complexity 
and  doubtless  has  made  hardly  more  than  a  beginning  at  the  present  time.  In 
fact,  it  has  thus  far  been  approached  only  by  confining  the  attention  to  a  very 
limited  number  of  special  functional  tj^T)es.'* 

*  "Sur  les  intdgrales  irregulieres  des  Equations  lin(5aires,"  Acta  Math.,  Vol.  8  (1886),  pp.  259- 
344.  Mention  should  be  made  also  of  Stieltjes  who  simultaneously  with  Poincare  resumed  the 
study  of  divergent  series,  confining  his  attention,  however,  to  the  computational  aspects  of  certain 
special  series.     (Thesis,  Ann.  de  I'Ec.  Nor.  (3),  Vol.  3  (1886),  p.  201.) 

'  For  the  more  accurate  statements,  see  Chap.  III. 

*  For  details,  see  Chap.  II. 


Preface  ix 

The  theory  of  summability,  or  second  general  aspect  of  divergent  series 
mentioned  above,  is  essentially  concerned  with  the  question  as  to  whether  in 
any  proper  sense  a  "  sum  "  may  be  assigned  to  the  series,  assumed  divergent, 

(3)  Han. 

n=Q 

This  question  has  been  scientifically  attacked  only  within  comparatively  recent 
years,  the  most  common  avenue  of  approach  being  through  the  so-called  boun- 
dary-value (Grenzwert)  problem  in  the  theory  of  analytic  functions.^  Thus 
Frobenius,  without  having  in  view  the  study  of  divergent  series,  showed  in  the 
first  place  that  if  one  has  a  power  series  whose  radius  of  convergence  is  equal  to  1 : 


(4)  XI  (inX'^ ',        ^  =  radius  of  convergence  =  1 

n=0 

and  writes  Sn  =  ao  -{-  ai  -\-  •••+««,  the 

00 

(5)  lim  HanX^  =  lim 


and  writes  Sn  =  ao  -{-  ai  -\-  •••+««,  then 

-^0  +  ^1  +    •  •  •   +  ^n 


a;=l— 0  n=0  m=oo  71  -p    i 

whenever  the  indicated  limit  on  the  right  exists.^  Now,  the  first  member  of  (5) 
is  naturally  associated  with  the  corresponding  series  (3)  (in  general  divergent) 
obtained  by  placing  x  =  1  in  (4).  Thus,  at  least  if  one  confines  the  attention  to 
divergent  series  (3)  of  the  particular  type  just  mentioned,  it  becomes  natural  to 
assign  sums  in  accordance  with  the  formula 

,.       5o  +  *1  +    •  •  •    +  Sn 

(6)  *  = !™  — ^+1       • 

whenever  the  indicated  limit  exists.  Moreover,  this  formula  finds  additional 
justification  in  the  demonstrable  fact  that  for  any  convergent  series  (3)  the  sum, 
regarded  in  the  ordinary  sense,  viz.,  s  =  lim  Sn,  agrees  with  that  given  by  (6) — 

w=oo 

i.  e.,  formula  (6)  is  consistent  Aside  from  this  one  formula  (6)  many  others  are 
now  known  which  serve  with  more  or  less  appropriateness  to  define  the  sum  of  a 
divergent  series,  both  when  the  series  is  of  the  special  type  above  mentioned  and 
when  otherwise.  To  what  ultimate  extent  these  formulas  are  appropriate,  how 
far  the  theories  of  summability  erected  upon  them  serve  any  justifiable  purpose 
in  analysis,  whether  the  different  sums  thus  assigned  involve  mutual  incon- 
sistencies— these  and  other  questions  may  well  be  asked  and  more  will  be  said  on 
this  point  presently.''     Suffice  it  to  say  here  that  formula  (6)  has  been  found  in 

6  For  an  elementary  description  of  the  problem,  see  Jahuaus,  "Das  Vcrhalten  der  Potenz- 
reihen  auf  dem  Konvergenzkreise  historisch-kritisch  dargestellt."  Program  des  Gymnasiums 
Ludwigshafen  (1901),  pp.  1-56.  See  also  Knopf,  "Grenzwerte  von  Reihen  bei  der  Annaherung 
an  die  Konvergenzgrenze."     Dissertation,  Berlin,  1907. 

6  "Ueber  die  Leibnitzschc  Reihe,"  Jour,  fur  Math.,  Vol.  89  (1880),  pp.  2G2-264. 

^  Interesting  comments  by  Pringsheim  relative  to  such  questions  are  to  be  found  in  Vol.  I 
of  the  "Encyklopiidie  der  math.  Wissenschaften,"  §§  39-40. 


X  Pkeface 

particular  to  yield  interesting  and  valuable  results  when  applied  to  Fourier  series 
and  the  other  important  allied  developments  in  mathematical  physics  —  develop- 
ments in  terms  of  Bessel  functions,  Legendre  functions,  etc.  Such  applications 
alone  go  far  toward  assuring  a  permanent  place  in  analysis  to  the  theory  of 
summability  as  now  commonly  understood. 

Turning  now  more  specifically  to  the  contents  of  the  present  volume.  Chapter  I 
considers  certain  aspects  of  the  so-called  Maclaurin  Sum-Formula,  the  especial 
aim  being  to  develop  and  summarize  into  actual  theorems  those  results  which 
are  of  importance  in  this  connection  to  the  study  of  divergent  series.  These 
when  once  obtained  are  of  particular  service  in  the  problem  of  determining  the 
asymptotic  developments  of  a  given  function,  and  it  is  to  this  that  Chapter  II 
is  then  devoted.  Beginning  with  very  easy  illustrative  studies,  the  Chapter 
proceeds  to  problems  of  greater  and  greater  difficulty  and  eventually  treats 
the  general  problem  already  considered  by  various  investigators  of  determining 
the  asymptotic  developments  of  the  general  integral  (entire)  function  of  rank  p 
(order  >  0),  following  which,  at  the  close  of  the  chapter,  the  problem  of  deter- 
mining the  asymptotic  developments  of  functions  defined  by  power  series  is 
briefly  considered.  Chapter  III  concerns  the  asymptotic  solutions  of  linear 
differential  equations  and  is  an  attempt  to  summarize  briefly  and  without  proof 
what  are  deemed  to  be  the  most  essential  results  thus  far  known  in  this  field, 
with  mention  also  of  the  corresponding  results  obtainable  in  the  study  of  linear 
difference  equations,  and  with  indications  as  to  certain  open  questions  still 
remaining  in  both  connections.  Chapter  IV  considers  the  theory  of  summability 
with  the  especial  attempt,  as  in  previous  chapters,  to  single  out  what  seems  most 
essential.  More  specifically,  it  makes  an  examination  of  a  few  of  the  standard 
definitions  of  "  sum  "  with  the  idea  of  subjecting  each  to  a  number  of  tests  which, 
as  the  author  has  come  to  view  the  subject,  every  such  definition  should  satisfy. 
For  example,  it  is  well  known  that  if  a  really  logical  general  theory  of  summa- 
bility is  ever  to  be  constructed  it  cannot  include  all  definitions  of  sum  that  satisfy 
merely  the  condition  of  consistency  (§  37)  since  this  alone  does  not  insure  unique- 
ness of  sum.  Therefore,  observing  the  genesis  of  the  whole  subject  from  the 
boundary  value  problem  as  described  above,  it  is  proposed  to  arbitrarily  limit 
the  general  theory  to  those  series  (3)  for  which  the  corresponding  power  series  (4) 
has  a  radius  of  convergence  equal  to  1  and  then  retain  only  such  definitions  of 
sum  as  give  the  unique  value 

s  =   lim  2_/  fln''^'". 

«  =  ]— 0  71=0 

Definitions  which  do  this  are  said  to  satisfy  the  boundary  value  condition  (§  39). 
Such  definitions  not  only  all  give  the  same  sum  to  a  given  series  (convergent  or 
divergent)  (3),  but  they  at  once  serve  a  useful  purpose  in  analysis  from  the  fact 
that  they  frequently  come  to  furnish  the  analytic  continuation  of  the  series  (4) 


Preface  xi 

over  some  portion  of  its  circle  of  convergence,  or  indeed  in  some  cases,  as  in  the 
definitions  of  Borel,  throughout  regions  lying  entirely  outside  that  circle.  How- 
ever limited  the  scope  of  a  general  theory  of  summability  as  thus  conceived,  it 
at  least  has  perfect  definiteness  and  logical  coherence  and  finds  immediate  use- 
fulness in  the  theory  of  functions  of  a  complex  variable,  and  we  venture  the 
opinion  that  some  such  characteristics  as  these  must  be  preserved  in  any  general 
theory  of  summability  that  is  to  retain  a  permanent  place  in  analysis.^  No 
attempt  will  be  made  here  to  describe  the  other  tests  which  Chapter  IV  sets  up, 
but  it  should  be  remarked  that  only  a  few  of  the  standard  definitions  of  sum 
are  tested  out  since  they  suffice  to  illustrate  the  spirit  of  the  undertaking.  The 
chapter  closes  with  a  brief  account  of  absolutely  summable  series  and  a  state- 
ment of  certain  supplementary  theorems  and  corollaries  upon  summability  in 
general. 

A  most  important  aspect  of  the  theory  of  summability,  as  the  author  regards 
it,  lies  in  its  applications  mentioned  above  to  Fourier  series  and  other  allied 
developments  in  mathematical  physics,  and  this  forms  the  subject  of  Chapter  V. 
For  the  sake  of  completeness  the  treatment  is  made  to  include  both  convergence 
and  summability.  It  is  based  upon  a  general  method  for  the  study  of  all  such 
developments  due  to  Dini  and  appearing,  though  in  somewhat  diffuse  and 
inaccessible  form,  in  his  great  work  entitled  "  Serie  di  Fourier  e  altre  rappre- 
sentazioni  analitiche  delle  funzioni  di  una  variabile  reale  "  (Pisa,  1880).  Dini 
naturally  considered  at  the  time  of  his  investigations  only  the  question  of  con- 
vergence (not  including  uniform  convergence),  but  his  methods  are  here  shown 
to  be  readily  extended  so  as  to  be  applicable  to  studies  in  summability.  Especial 
effort  has  been  made  here  as  in  the  other  chapters  to  summarize  all  essential 
conclusions  from  time  to  time  into  actual  theorems. 

To  Professor  Alexander  Ziwet  the  author  would  here  express  his  deep  grati- 
tude. Not  only  has  the  book  enjoyed  the  benefits  of  his  critical  judgment  in 
many  ways,  but  his  sympathy  and  kindly  interest  have  served  as  a  constant  en- 
couragement, and  indeed  they  are  responsible  in  no  small  measure  for  the  ap- 
pearance of  the  whole  in  its  present  form.  The  author  is  much  indebted  also 
to  his  colleagues  Professors  C.  E.  Love  and  Tomlinson  Fort,  the  former  for 
various  suggestions  and  criticisms,  and  the  latter  for  the  valuable  aid  he  has 
rendered  in  reading  the  proofs. 

Ann  Arbor, 
AprU,  1915 

8  The  adoption  of  any  one  definition  for  "summable  series"  evidently  involves  the  exoludinp; 
of  many  scries  previously  classed  as  summable;  yet  we  believe  the  time  has  arrived  when  a  single 
universal  definition  should  if  possible  be  agreed  upon,  however  disastrous  its  immediate  efi'ects 
upon  one  or  more  of  the  special  forms  of  definition  now  current.  The  present  situation  in  this 
matter  is  strikingly  analogous  to  the  state  of  confusion  which  led  Cauchy  and  Abel  to  the  formu- 
lation of  their  universal  definition  for  "convergent  series,"  notwithstanding  the  exclusions  brought 
about  and  the  consequent  objections  urged  by  contemporary  mathematicians. 


CHAPTER  I 

THE  MACLAURIN   SUM-FORMULA,  WITH  INTRODUCTION  TO  THE   STUDY  OF 

ASYMPTOTIC   SERIES 

1.  The  following  formula  (Maclaurin  Sum-Formula^) 
Efix)  =  i   rf{x)dx  -  i  [/(6)  -  /(a)]  +  ^.f  [fib)  -  fia)] 

+  •  •  • ;         Bm  =  inih  Bernoulli  number 

plays  an  important  part  in  the  modern  theory  of  divergent  series  and  we  shall 
therefore  begin  by  pointing  out  certain  facts  (cf.  Theorems  I,  II,  III  and  IV) 
connected  with  its  legitimate  use.  These  will  form  the  basis  of  the  studies 
undertaken  in  Chapter  II. 

Following  the  discussion  of  (1),  we  shall  also  give  in  the  present  Chapter 
(cf.  §§  13-17)  an  outline  of  the  general  theory  of  asymptotic  series  as  originally 
developed  by  Poincare  in  his  classical  memoir  in  the  Acta  Mathematica  (1886), 
the  elements  of  this  theory  being  likewise  needed  for  the  proper  development 
of  Chapter  II. 

2.  In  order  to  carry  out  the  desired  studies  relative  to  the  formula  (1),  let 
us  begin  by  supposing  that  there  is  given  any  function  Ux  (real  or  complex)  of 
the  real  variable  x  which,  together  with  its  first  2m-j-l  derivatives,  is  continuous 
within  a  certain  interval  (a,  b) .  For  any  value  of  x  such  that  a  ^  x  ^  x  -\-  h  <  h 
(h  =  constant)  we  may  then  write 


2!    "    ^         ^(2m)! 


Aux  =  Ux+h  —  Ux  =  hux  +  w,  Ux"  +  •  •  •  +  yc,_Vi  Ux^^""^ 


^^^  ^"  (h  -  z) 


^  i       (2m)  1     '''^'    ^'^' 


as  appears  directly  upon  applying  an  integration  by  parts  2m  times  to  the  last 
term  in  the  second  member. 

More  generally,  it  appears  in  like  manner  that  when  0  ^  ^•  ^  2m  —  1  we 
may  write 

^  Known  also  as  the  Euler  sum-formula.     For  comments  upon  the  historical  aspect  of  the 
subject,  see  Barnes,  Proceedings  London  Math.  Soc.  (2),  Vol.  3  (1905),  p.  253. 
2  1 


2  The  Maclatirin  Sum-Formula 

12  I2m-fc 

while  the  corresponding  formula  for  the  case  k  =  27?i  is 
(4)  Awx^""^  =  J  wf™+^Wz. 

Whence  if  //o,  i^i,  H2,  •  •  •  ^2m  be  the  2m  +  1  constants  determined  by  the 
equations 

Ho  =   1,  H2m  =  0, 


(5) 

we  shall  have 


or 

2m 

(6)  hi,'  =  E  Hj^h^i^u,^^^  +  r^(.r,  A) 
where 

(7)  rirK)  =  -    f  ,(--)  V  ^^^^^^^^^^^' ^. 
(7)  r.(.r, /»)-       j^   ^^.+.     2^^       {2m  -  k)\      '^^' 

Formula  (6)  bears  a  close  relation,  as  we  shall  see,  to  the  Maclaurin  Sum- 
Formula  (1). 

We  first  proceed  to  determine  the  values  of  the  constants  Hu,  noting  certain 
changes  which  thereby  become  possible  in  the  form  of  (7). 

If  we  place 

'^""^"'^  ~  (2m)!"*"  (2m-  1)^    (2m  -  2)r    (2m  -  4)!"^  "  * 
(8) 

77„       „^2m-2~2 

and 

(9)         4^uz)  =  (^^,^^3)-,+  (o^^T^T^!"^  •  •  •  "^  — rr~ 

we  have 


.(.r,  h)  =  -    f  u^:!:t'\^2m(h  -z)  +  hUh  - 
Jo 


z)]dz. 


Let  us  now  develop  (p2m{h  —  2)  +  \l/2m{h  —  z)  in  ascending  powers  of  z.     We 
obtain 


General  Theorems  3 

(p2m(.h  —  2)  +  \l/2m(h  —  z)    =   J2  Hk  —7^ -TTj- 

But  from  (5)  we  have 

2m-j  TT  1  JJ 

E  {2m- k-j)r'^ +"'--' ="'--'     0-  +  2™-l);        E^j^^  =-//,. 
Whence, 

<P27n{h  —  z)  +  \p2mih  —  z)  =  7^^;+  (2^  —  1) !  "^  ?J  *^~'  ■^^'^•^-'"-^■~jl 

=    <P2miz)   —   \p2miz). 

Thus,  if  we  place  z  =  A/2  we  obtain 

l/'2m  (    9  j  =    -   lAam  (   2    )  '  ^2m  (    2    )  ^    ^• 

The  last  relation,  however,  cannot  exist  for  all  positive  integral  values  of  m  unless 
the  coefficients  of  the  various  terms  of  4^2m{z)  are  each  equal  to  zero.     Hence, 

(10)  Hz  =  H,==  Ht=  ■■'  =  H.,m-i  =  0, 
and  we  obtain  the  relations 

(11)  rUx,  h)  =  -    f  v':-:t''<P2m{h  -  z)dz, 

(12)  iP2m{ll  —  Z)    =    (p2m{z). 

As  to  the  coefficients  Hi,  H2,  Hi,  Fh,  •  •  •  H2m-2,  we  have^ 

(13)  Hi=  -I        H2r  =  ^  (2r)!     ^'■'         ^  =  1.  2,  3,  •  •  •  (m  -  1) 

where  Br  is  the  rth  Bernoulli  number. 

3.  We  now  proceed  to  establish  the  two  following  properties  of  the  functions 
(P2m{z)  (commonly  known  in  case  //  =  1  as  the  functions  of  Bernoulli) : 

(a)  "  The  function  (p2m{z)  does  not  change  sign  between  z  =  0  and  z  =  h  and 
is  positive  in  this  domain  when  m  is  even  and  negative  when  m  is  odd." 

(6)  "  The  expression  \ip2miz)  \  when  considered  for  values  of  z  between  s  =  0 
and  z  =  h  has  its  maximum  value  at  z  =  h/2." 

^  This  result  like  others  which  concern  the  well-known  properties  of  the  Bernoulli  numbers 
and  functions,  will  here  be  presupposed.  For  a  proof,  see  Malmsten,  Journ.  fiir  Math.,  Vol.  35 
(1847),  p.  64. 


4  The  IMaclauein  Sum-Formula 

For  the  proof  of  (a)  let  us  consider  the  expression 

^      ^2m-3  Hihz'"'-'        Hjhh''"'-^        Hi¥f^ 

<p'2m-i{z)  -  ^2^^  _  3)  I  +  (2m  -  4)  r    (2m  -  5) !  "^  (2m  -  7) !  "^  "  * 

Supposing  for  the  moment  that  this  is  positive  whenever  0  <  2  <  A/2,  let  us 
multiply  it  by  hr~"'dh  and  integrate  from  h  =  h  to  A  =  +  00 .  The  result, 
except  for  the  factor  h~^"^'^^,  is 

1"    /o_  C\  !    /O™  ON      I 


(2m  -  3)  1  (2m  -  1)   '    (2m  -  4) !  (2m  -  2)   '    (2m  -  5) !  (2m  -  3) 

"^         3!  5         "*~        1!3 

and  this  must  likewise  be  positive  when  0  <  2  <  A/2.     Let  us  now  multiply 
the  last  expression  by  dz  and  integrate  from  2  =  2  to  2  =  A/2.     We  obtain 


^-//,„..;,-T'^-^'  +  ' 


2"'U/ 


But  (p'2„Xh/2)  =  0,  as  follows  from  (12).  We  therefore  conclude  that  if  (P2,n-2(z) 
is  positive  when  0  <  2  <  A/2,  then  ^2,,, (2)  is  negative  throughout  the  same 
domain.  Now,  ^2(2)  =  2-/2  —  2A/2,  (p',{z)  =  2  —  A/2.  Whence,  (p'o.Sz),  con- 
sidered for  values  of  2  within  the  indicated  interval,  will  be  positive  or  negative 
according  as  m  is  even  or  odd,  while  the  expression 


<p2m(z) 


=    I      <Ptm.'{z)dz 
Jo 


will  be  positive  if  m  is  even  and  negative  if  m  is  odd.  It  follows  from  (12)  that 
<P2m{z)  has  the  indicated  properties  for  the  interval  0  <  2  <  A. 

Concerning  (b)  we  note  that  if  m  is  even  we  have  shown  that  ip^miz)  is  positive 
when  0  <  2  <  A/2  and  that  <p2^{hf2)  =  0.     Moreover,  since 

(pLXz)  =  -  <p'.„Xh  -  2), 

the  same  function  is  negative  when  A/2  <  2  <  A.  Thus,  the  statement  in 
question  follows  from  elementary  considerations  in  the  theory  of  maxima  and 
minima.  Likewise  we  reach  the  same  result  when  m  is  odd,  since  (f'zmiz)  is  then 
negative  from  2  =  0  to  2  =  A/2  and  positive  from  2  =  A/2  to  2  =  A. 


General  Theorems  5 

4.  These  results  being  established,  we  return  to  formula  (6).     In  this  formula 
let  us  take 


i(x  =  I    f{x)dx, 

fJa 


where  f{x)  together  with  its  first  2m  derivatives  is  continuous  from  x  =  a  to 
X  =  h.  Then  w^  together  with  its  first  2m  +  1  derivatives  will  be  continuous 
within  the  same  interval  so  that  for  any  value  of  x  for  which  a^x<x-\-h^b 
we  shall  have  (cf.  (5),  (11),  (13)) 

hm  =  r'mdx  -  I  [/(.r  +  h)  -  f{x)]  +  -^  [fix  +h)-  fix)] 

(14)  .       -  ~f  [fix  +h)-  fix)]  +  .  • .        - 

+       (2m  -2)!       [/^''""'^(•^-  +  ^^  -  /^''""'H^)]  +  r.(.T,  A), 
where 

(15)  rr,ix,  h)=^  -  f  f^Kx  +  z)<p2miz)dz. 

Let  us  now  suppose  that  6  —  a  is  an  integral  multiple  of  h,  i.  e.,  b  —  a  =  nk 
and  allow  x  to  take  successively  the  values  a,  a  -\-  h,  a  -\-  2h,  •  •  •  a  -{-  (71  —  l)h. 
By  adding  the  corresponding  results  (14)  and  dividing  by  h  we  obtain 

E/(«  +  9/0  =  Zfix)  =  \  ffix)dx  -  i  [fib)  -  fia)] 

5=0  x=a  "  •/« 

(16)  +  Y[  U'ib)  -  fia)]  -  ^  [fib)  -  fia)]  +..•  +  ... 

+        iL-2)l       f/'"'"''^^)  -  /^'"'~'^(«)]  +  ^-' 
where 

(17)  /?„,=    _  i    r    Zf^Kx  +  Z)<p2n,iz)dz. 

By  placing  m  =  co    ^ve  thus  arrive  at  formula  (1)  provided,  however,  that 

lim  R„,  =  0.3 

7n=oo 

5.  We  proceed  to  consider  certain  properties  of  the  remainder  Rm  correspond- 
ing to  the  cases  in  which  fix)  is  real.     From  result  (a)  of  §  3  we  may  apply  the 

^  For  noteworthy  cases  in  which  this  condition  ia  fulfilled,  see  Markoff's  "  DLfferenzen- 
rechnung  "  (Leipzig,  1896),  Chap.  9,  §  8. 


6  The  Maclaurin  Sum-Formula 

first  law  of  the  mean  for  integrals  and  write 

7C  =  ^E  f  H.^-  +  eh)   (    cp2miz)dz;        0  <  ^  <  1 

"■      z=a  Jo 

or,  since^ 

(18)  -  I   <pUz)dz  =  j^, 

we  shall  have 

Whence,  also 

_r.Bmh^  f-l<0<l 

(20)  i4.-e    ^2^^^^,    (6-a)Ji;  |  M  >  |/2-(a:)  | ;     a^x^b, 

so  that  we  reach  in  summary  the  following  result: 

"  If  f{x)  be  any  (real)  function  of  the  real  variable  x  which  together  with 
its  first  2m  derivatives  is  continuous  within  the  interval  (a,  b)  we  may  write 
formula  (16)  in  which,  if  M  represents  a  value  as  great  as  the  maximum  value  of 
|/(-'"^(a;)|  within  the  same  interval,  the  expression  Rm  satisfies  relations  (19) 
and  (20)." 

6.  Other  important  forms  for  the  remainder  in  the  Maclaurin  sum  formula 
may  be  obtained  when  further  hypotheses  are  placed  upon  f{x).  Thus,  let  us 
suppose  in  the  first  place  that /^-'"^  (.r)  does  not  change  sign  between  x  =  a  and 
X  =  b.     By  applying  the  first  law  of  the  mean  for  integrals  we  may  then  write 

Rm  =  ^<P2m{eh)    f'j:P'"\x  +  Z)dz 

(21) 


^  <P2m{dh)[P"^-'Kb)  -  f'^-'Ka)];         0  <  0  <  1. 


h 
Whence,  by  (a)  and  (b)  of  §  3, 

(22)  Rm  =  ^e^2m  (J)  [P-"Kb)  -  p-^-'Ka)];        0  <  0  <  1. 

Moreover,  from  (8)  and  (13)  we  have 

<P2my2)-  "'     I  (2m) !  2^'"      2  (2m  -  1) !  22'"-i  ^1  •  2  (2to  -  2) !  2^""-^ 

4!  (2m- 4)1  22'"-^^         ^^      ^    (2m- 2)!  2!  2  J 

and  it  is  a  demonstrable  property  of  the  Bernoulli  numbers  that  the  expression 

^  See  Malmsten  (l.  c),  p.  64,  4. 

^  Due  originally  to  Poisson.     See  Mem.  de  I'Acad.  des  Sciences,  Vol.  6  (1823),  p.  590. 


General  Theorems 
here  appearing  in  square  brackets  is  equal  to 

(23)  (      1)      22"i-i     (2m)!' 

Whence,  by  adding  and  subtracting  the  term 

(—    n'"+17?    /j2m-l 


(2m) 


in  the  second  member  of  (16)  we  obtain  the  following  result: 

"  If  /(.r)  be  a  (real)  function  of  the  real  variable  x  which  together  with  its 
first  2m  derivatives  is  continuous  within  the  interval  {a,  h)  and  if  its  2mth  deriva- 
tive does  not  change  sign  between  the  same  limits,  we  may  write 

Zfix)  =  \  I    fix)dx  -  \  im  -  /(a)]  +  ^  [f  (6)  -  f  (a)]  -  ^ 

x=a  '^  *Ju  *^ ' 

where 

ft,=  (-  i)"+'(e^9=^-  O^St'-'''"""^''^  -/'^"-"Wl;      0  <  e  <  i." 

Since 

22m  1  2^"*  1 

(24)  0  <  Q    2^"^-^    "  ^^      2^"-     ^  ^ 

we  see  that  under  the  hypotheses  of  the  above  result  the  series  (1),  even  though 
it  be  divergent,  may  be  used  to  compute  the  value  of 


h-h 


(25)  llj{x) 

x=a 

with  an  error  numerically  less  than  the  absolute  value  of  the  last  term  taken. 

More  generally,  it  appears  in  the  same  manner  that  we  shall  have  the  above 
result  whenever /(.r),/'(.r),/"(.T),  •  •  •  p'^Kx)  are  continuous  within  the  interval 
(a,  h),  while  the  expression 

llP"'\x  +  z) 

x=a 

does  not  change  sign  between  2=0  and  z  =  h. 

7.  Again,  let  us  now  suppose  that  neither  of  the  expressions 


h-h  1>-h 


(26)  HP'^Hx  +  z),  Zp-'+'Kx  +  z) 

x=a  ■r=a 

changes  sign  between  z  =  0  and  z  =  h.  Replacing  m  by  m  +  1  in  (16),  using 
8  See  Malmsten  {I.  c),  P-  70. 


8  The  INlACLAimm  Sum-Formula 

therein  the  form  for  i?mfi  determined  by  (19),  and  comparing  the  result  with 
that  of  §  5  (in  which  m  is  left  unaltered),  we  obtain 

D  7,2m+2  h-h  I      o2m  _   1  ID     Z,2m-1 

But 

/th  b—h 
/(2'»-l)(fe)   _/(2m-l)(a)   =     I       J2P"'KX+Z)dz. 

Whence,  upon  recalling  that  Bn  and  Bm+i  are  both  positive,  we  see  that  the 
expression 

22m  1 

"      22m— 1  ■'■ 

will  be  negative  and  numerically  less  than  1  in  case  expressions  (26)  are  of  the 
same  sign  between  z  =  0  and  z  =  h,  while  it  will  be  positive  and  no  greater  than 

22"»  _  1  22'"-i  —  1 

22m— 1  -'■    ~~         22™~1 

in  case  expressions  (26)  are  of  opposite  sign  throughout  the  same  domain.     Thus 
we  reach  the  following  result : 

"  Let  f(x)  be  a  (real)  function  of  the  real  variable  x  which  together  with  its 
first  2m  derivatives  is  continuous  within  the  interval  (a,  b)  and  is  such  that 
neither  of  the  expressions 

ZP'^Kx  +  z),      ZF-'+'Kx  +  z) 

x=a  z=a 

changes  sign  between  2=0  and  z  =  h.     Then,  according  as  these  expressions 
preserve  the  same  or  opposite  signs  for  the  indicated  values  of  z,  we  may  write 

E/Or)  =  \  fmdx  -  \  \m  -  /(a)]  +  ^  [/'(&)  -  /'(a)] 


(27)  --^  [/'"(&) -/'"(«)]+  ••• 

+  ^"  ^^'"  (2m -2)!  [/'"""''(^)  -  F-'^-'^i^)]  +  Rn.> 
where 

B    ^2m-l 

R^=  i-  ir+^e  -g^  [/(2-i)(6)  _  /(2-i)(a)];        0  <  e  <  1 
and 

EV)  =  \  fmdx  -  i  [/(6)  -  /(a)]  +  1^  [j\h)  -  f  (a)] 

(28)  -^' [/'"(&) -/'"(«)]+  ••• 


General  Theorems  9 

where 

02m—l   ID     Jj2m—1 

R,n  =  (-  i)-+^e    ^,^_,    ^^  [f'-Hh)  -  r---Ha)];      0  <  e  <  1." 

Formula  (27)  was  first  established  by  Jacobi'^  in  1834.  Whenever  the  con- 
ditions for  its  use  are  satisfied  it  is  seen  that  the  sum  of  any  number  of  terms  in 
the  series  (1)  (convergent  or  divergent)  gives  the  value  of  (25)  with  an  error  having 
the  same  sign  as  that  of  the  first  term-  neglected  and  less  numerically  than  the 
absolute  value  of  that  term.  Formula  (28)  is  due  to  Malmsten.^  Whenever 
it  may  be  used  the  sum  of  any  number  of  terms  in  the  series  (1)  gives  the  value  of 
(25)  with  an  error  having  the  same  sign  as  that  of  the  last  term  taken  and  less 
numerically  than  the  absolute  value  of  that  term. 

8.  Another  important  and  well-known  form  for  the  remainder  in  the  Mac- 
laurin  Sum-Formula  may  be  obtained  when  the  function  f{x)  may  be  regarded 
as  an  analytic  function  of  a  complex  variable. 

To  see  this  we  recall  in  the  first  place  that  if /(w)  and  ^(w)  are  any  two  func- 
tions of  the  complex  variable  w  {w  =  x  -\-  iy)  both  analytic  and  single-valued 
in  the  neighborhood  of  the  point  w  =  a  and  of  which  the  second  has  a  zero 
of  the  first  order  at  the  same  point,  then  we  have  the  formula 

J   <Piw)  ip{a)  [77,  ^  0     if    6  =  27r, 

where  the  integration  is  taken  in  the  positive  sense  along  the  arc  of  a  circular 
sector  of  small  radius  e  and  center  at  to  =  a  and  whose  angle  is  6.  In  fact,  this 
formula  results  directly  from  well-known  principles  in  the  theory  of  complex 
integrals  upon  observing  that  in  the  present  instance  we  may  develop  the  func- 
tion f{w)/(p{iv)  in  the  form 

-t-  p{iv) ;        c_i  = 


w  —  a      ^^   ■"  (p'(a) 

where  p{iv)  is  analytic  at  the  point  w  =  a. 

An  immediate  and  useful  corollary  of  (29)  is  as  follows: 

"  If  f{w)  and  (p(w)  are  any  two  functions  of  the  complex  variable  «'  both  of 
which  are  single-valued  and  analytic  in  a  region  A  of  the  ?t'-plane  and  of  which 
the  latter  vanishes  within  A  only  at  the  points  iv  =  Xi,  Xo,  •  •  •  X„  which  are 
zeros  of  the  first  order,  and  if  Cn  designate  any  contour  lying  within  .1  and 
including  the  points  iv  =  Xi,  X2,  •  •  •  Xn,  we  shall  have 

where  the  indicated  integration  is  performed  in  the  positive  sense." 

T  See  Journ.  fiir  Math.,  Vol.  13  (1834),  p.  270. 
'  See  Malmsten  {I.  c),  p.  72. 


10 


The  Maclauein  Sum-Formula 


We  proceed  to  apply  formulas  (29)  and  (30)  to  our  present  problem.^     For 
this  purpose  let  us  take^^ 

(31)  ip{ic)  =  gC^'iM) ("'-«)  -  1 

and  let  us  suppose /(w)  to  be  any  function  which  is  analytic  throughout  a  vertical 
strip  of  the  w-p\a,ne  extending  to  an  infinite  distance  both  above  and  below  the 
axis  of  reals  and  including  the  two  real  points  w  =  a,  iv  =  b  (b  >  a).  For  the 
contour  C„  let  us  take  that  formed  by  the  line  w  =  a  -\-  iy  (the  point  lo  =  a 
excluded),  by  the  line  w  =  h  -\-  iy  {w  =  h  excluded)  and  by  the  lines  id  =  x  ±  ij 
(j  =  constant  >  0)  together  with  small  semicircles  of  radius  e  >  h  about  the 
points  IV  =  a,  20  =  h,  the  former  extending  to  the  right  and  the  latter  to  the 
left. 

Since  ^(w)  has  zeros  of  the  first  order  at  the  points  iv  =  a  -\-  jih;  y  =  0, 
1,  2,  •  •  •,  while  at  the  same  points  (p'{w)  =  2Tri/h,  we  obtain  as  a  result  of  (30) 


(32) 


hZfix)  =  hf{a)+  f 


fiw) 


div. 


We  proceed  to  study  in  further  detail  the  complex  integral  here  appearing. 


Fig.  1 
First,  the  contribution  coming  from  the  side  J  A  (see  Fig.  1)  is 

'7(^ 


•-r 


ij)  , 
T7  ax, 


ip{x  -  ij) 

and  since  (p{x  —  ij)  becomes  infinite  when  j  =  +  oo  like  e-"-''"',  we  have  but  to 
suppose  that/(w)  satisfies  the  following  supplementary  condition: 


(33) 


lim  fix  -  ij)e--''^l''  =  0; 


x^b 


in  order  to  have  7i  =  0  provided  we  take  j  =  oo.  In  particular,  condition  (33) 
will  be  satisfied  whenever  \f{w)  \  remains  less  than  a  constant  for  all  values 
of  w  within  the  strip  already  mentioned. 

'  See  Petersen's  "  Vorlesungen  uber  Funktionstheorie  "  (Copenhagen,  1898),  pp.  161-169. 
'°  It  is  to  be  understood  that  the  constants  a,  b,  h  have  the  meanings  already  introduced; 
viz.,  h  =  a  +  7ih;  h  >  0,  n  =  positive  integer. 


General  Theorems  11 

Secondly,  let  us  consider  the  contribution  coming  from  the  portion  DEFG. 
By  writing 

(p{w)  <p{w)  ''^    ' 

and  observing  that  the  integral  oiji^w)  over  DEFG  is  equal  to  that  over  DCHG, 
the  contribution  in  question  becomes 

»J, ^(^rni) '^'■'+'i ^w+M ^y 

Of  the  integrals  here  appearing  we  observe  that  the  third  may  be  neglected  by 
taking  j  =  +  °°  provided  that  f(iv)  satisfy  the  following  supplementary  con- 
dition: 

(34)  lim  fix  +  ij)e-'-''^l''  =  0;        a  ^  x  ^  b. 

Next,  the  contributions  from  the  semicircular  arcs  BCD  and  GHI  are  equal 
respectively  to  —  {hj2)f{h)  and  —  (h/2)f{a)  except  for  expressions  that  become 
infinitesimal  with  e,  as  follows  from  (29). 

We  shall  now  assume  not  only  the  existence  of  (33)  and  (34)  but  that  of  the 
following  stronger  condition: 

(35)  lim  fix  ±  ij)e^'>-^''"'^'  =  0;        a^x^h, 

where  t]  is  an  assignable  positive  quantity.  If  we  then  take  account  of  the  two 
remaining  contributions,  viz.,  those  arising  from  the  sides  AB  and  IJ ,  we  obtain 
in  summary 

h  1^  fix)  =  fiw)dw  -  r  [fib)  -  fia)]  +  i        ,        .   dij 

x=a  Jghcjd  ^  Ji  <P\P  -r  W) 

.  r  hja  +  in)  +  ma  +  iy)  ,   ,  ■  f-'fC'  +  iy)  , 

.r-«/(a+M, 

+  'J_,    via  +  iy)'^^' 

in  which  the  various  improper  integrals  have  a  meaning  by  virtue  of  (35). 
Let  us  next  allow  e  to  approach  zero.     Since 

<pia  +  iy)  =  ifib  +  iy)  =  e-~''y'^  -  1, 
we  thus  arrive  at  the  equation 


12  The  IMaclaurin  Sum-Formula 


(36) 


h  ZKx)  =    f  f{x)dx  -  I  [m  -  f{a)] 


+ 


1   r[Kx  +  iy)-f{x-iy)]lz:i, 


Moreover,  the  function  (1/i)  [/(.t  +  iy)  —  f(x  —  iy)Yj^^z'',,  being  real  when  y 
is  real,  may  be  expanded  by  Taylor's  formula  (with  remainder)  into  the  form 


ihf 


(x)y  -  lr{x)y'  +  •  •  •  +  (L-"'l)V'''""''^^'^^'"'~'  Tl 


Recalling  finally  that^^ 

we  reach  the  following  result: 

"  If  the  function  f{iv)  is  analytic  throughout  a  vertical  strip  of  the  lo  complex 
plane  extending  to  an  infinite  distance  both  above  and  below  the  axis  of  reals 
and  including  the  real  points  iv  =  a,  iv  =  h,  and  is  furthermore  such  that 

lim  f{x  ±  iy)e-'^--'"^^'y  =0;        a^x^h, 

y=+oo 

where  rj  is  some  assignable  positive  quantity,  we  may  write 

2  /(.r)  =  \  f  Kx)dx  -  h  im  -  /(a)]  +  ^'f  I  fib)  -  f{a)] 

B2h^ 

-^[rib)-r{a)]+  ••• 


where 


lin 


(-l)"    r  [P'^Kx  +  idy)  -  P^Kx  -idy)]-l 
{2m)\ihX  g2,w;._  1  y    (^y 

6=1     when    m  =  0, 

0  <  ^  <  1     lohen     m  =  1,  2,  3,--- 

Equation  (36)  with  A  =  1,  was  first  given^-  b}^  Plana  in  1820  and  soon  after- 
wards by  Abel.^^  The  same  result  was  obtained  through  the  calculus  of  residues 
for  the  first  time  by  Kronecker^^  in  1889. 

"  See  Malmsten  {I.  c),  p.  59. 

"  See  Mem.  della  Accad.  delle  sci.  di  Torino,  Vol.  25  (1820). 

i»  See  Oeuvres  completes  (1881),  Vol.  1,  pp.  21-25.     Ibid.,  pp.  34-39. 

"  See  Journ.  fiir  Math.,  Vol.  105  (1889),  p.  354. 


General  Theorems  13 

9.  We  proceed  to  note  certain  theorems  which  follow  from  the  preceding 
results  and  which  will  prove  useful  in  the  study  of  divergent  series  (Chap.  II). 

Theorem  I.  Letf{x)  be  any  (real)  function  of  the  real  variable  x  which  together 
with  its  first  2m  derivatives  is  continuous  throughout  the  infinite  interval  x  >  a. 
Also,  lei  it  be  supposed  that  the  following  series  is  convergent-}" 

(37)  E  r  f^'^'Ky  +  t)<P2jt)dt, 

in  ivhich  ipim{t)  represents  the  2mth  Bernoulli  function.     We  may  then  write 

Z  /(^)  =  C^+  ^/(.^•)c/x  -  hm  +  f;  fix)  -  §  f"'{x)  +  •  •  • 

x=a  ^a  ^  ' 

ichere 

^m{x)  =  Z      f'-^Ky  +  0^2.(0^^^  =  ^    J  ^,  " Hf'-Ky  4-  dy)- 

^^^^  \x^a 

\0<dy<\, 

and  ichere  Cm  is  a  constant  as  regards  x,  defined  by  the  equation 

C^  =  l/(a)  -  fy/'(«)  +  4f/'"(«) +     (2Jr^- off  f ''""''' ^""^  -  ""•(«)• 

In  fact,  the  expression  fi^(.r)  will  exist  for  .r  =  a,  a  +  1,  a  +  2,  •  •  •,  and  by 
the  results  of  §  4  we  shall  have 

Jlfix)  =    rf{x)dx  -  i  [fix)  -  /(a)]  +  |y'  [fix)  -  fia)] 

x=a  tJa  ~  ' 


where 


+   i2m-2)l^^''^~''^'^^  ~  /(^-^>(a)]  +  R„ 


/»1  x-l 

IL=    -  Z/^""^^/  +  t)^im{t)dt  =    Qmix)   -   Qmia). 

Jo      y=a 


But  this  result  is  coextensive  with  that  indicated  in  the  theorem.  As  to  the 
second  form  there  given  for  fim(a;),  we  observe  that  by  virtue  of  statement  (a) 
of  §  3  we  may  apply  the  first  law  of  the  mean  for  integrals  to  each  term  of  the 
series  representing  fi,„(a-),  thus  writing 

^ix)  =  Zf^'^'Hy  +  Oy)  f  <P2,nit)dt;        0  <  ^,  <  1, 

1^  It  will  be  understood  that  in  this  and  the  following  two  theorems  y  takes  onlj*  the  values 
a,  a  +  1,  a  +  2,  •  •  • . 


14  The  Maclaurin  Sum-Formula 

which,  upon  using  (18),  becomes 

(39)  an(.r)  =  ^7^^'  HF'-Hy  +  By). 

We  add  that  in  case  lira /(^p-d (.r)  =  0;  p  =  1,  2,  3,   •  •  •,  the  constant  Cm 

x=oo 

will  be  independent  of  m  as  well  as  of  x,  for  we  shall  then  have 

SO  that  by  placing  x  =  co   and  observing  that  lim  Oi(a:)  =  lim  fi,„(.r)  =  0,  we 
obtain  Ci  =  Cm. 

Theore:m  II.  Let  f(x)  be  any  (real)  function  of  the  real  variable  x  ichich 
together  icith  its  first  2m  derivatives  is  continuous  throughout  the  infinite  interval 
X  >  a.  Also,  let  it  be  supposed  that  f^-^^{x)  does  not  change  sign  within  the  same 
interval  and  that  lim /^-"""^^ (.r)  =  0.     We  may  then  ivrite 


Zm  =  Cm  +  ffi'-^-Hv  -  hfi'V)  +  |jV'(aO  -  ^f'ix)  + 

(2m) 


where 


^'^  ,e,„(.T)r-^>(.r);        ^•''^'' 


(2m)  r'"^-"^-'  ^-^^^  [-  l^e^(a:)^l, 

and  where  Cm  is  a  constant  as  regards  x,  defined  by  the  equation 

Cm  =  hKa)  -  -^^f'ia)  +  -^r(a) +     (2^)1  "/^^"""(«)  "  "-(«)• 

To  prove  this  theorem  we  first  observe  that,  as  a  result  of  our  hypotheses 
upon /^-'"^ (a:),  the  terms  of  the  expression 

Fm{b,  X)    =    E     f  f^'^'Ky  +   t)ip2m{t)dt;  b  >   X 

y=x  I/O 

will  all  have  the  same  sign,  so  that  Fm{b,  x)  is  either  an  ever  increasing  or  an 
ever  decreasing  function  when  b  increases;  also  by  treating  Fm{b,  x)  as  we  did 
the  Rm  of  (17)  by  means  of  (21),  (22)  and  (23)  we  obtain 

^-^^'   •''^    =  (2m)  l'"  -^rrT-r  Vm{b,  x)  {/<^'"-^>(6)    "  f^'-'\x)  }  J 

0  <  vmib,  x)  <  1.1^ 
"  The  expression  G  appearing  in  §  5  is  in  general  a  function  of  m,  a,  b  and  h.     Since,  in  the 
present  instance  we  have  h  =  1,  a  =  x  we  represent  9  by  ??m(6,  x). 


General  Theorems  15 

Whence,  the  expression 

Fm{x)  =  Hm  Fm{h,  x) 

exists,  and  since  by  hypothesis 

limj(2m-l)(5)    =    0, 
6=00 

we  shall  have 

0  ^  Vmix)    ^   1 

or 


(40) 


(_  n^+iD    f  o-"*  —  1         1 


Thus,  fim(-^)  exists  and  has  the  form  indicated  in  the  theorem. 
Now,  by  equation  (16)  we  shall  have  also 

E /(.!•)  =  fj{x)dx  -  i  [j{:x)  -  f{a)]  +  ^  [f  (.t)  -  f  (a)] 

+       (2m)!    "[Z^'""'^^-^)  -P'^-'Ka)]  +  0.(:r)  -  i2.(a). 

Thus,  we  reach  the  desired  result.  Again  we  note  that  Cm  will  be  independent 
of  m  as  well  as  of  x  whenever 

lini/(2p-i)(^)  =  0;        2^  =  1,2,3,  •••. 

«=00 

Theorem  III.  Let  f{x)  he  any  {real)  function  of  the  real  variable  x  which 
together  with  its  first  2m  +  2  derivatives  is  continuous  throughout  the  infinite  interval 
X  >  a.  Also,  let  it  be  supposed  thatf^^'^\x)  and  f^^"^'^^\x)  do  not  change  sign  within 
the  same  interval,  ivhile 

\\mf^'^-'\x)  =  0;        p=  1,  2,  3,  •... 

a;=oo 

Then,  iff^^'^^x)  and  f^^"''^^'' (x)  preserve  the  same  sign  (x  >  a)  ice  may  write 
Zm  =  C  +   rfix)dx  -  y{x)  +  §lf'{x)  -  ^f"'{x)  +  • . . 

ivhere 

^mix)   =    E     rf''-\y+t)<p2mmt  =    (-    ir^'en.ix)j-^~-.f^'--'\x)-, 
y=x  Jo  \^m)  i 

X  ^  a 

0  ^  Omix)  ^  1, 

and  where  C  is  a  constant  as  regards  both  m  and  x,  defined  by  the  equation 


16  The  Maclaurin  Sum-Formula 

On  the  other  hand,  iff^'^'"^(x)  and  f^-"''^^\x)  preserve  opposite  signs  (x  >  a)  {other 
conditions  remaining  as  before)  we  may  write 

%m  =  C  +  £f{x)dx  -  I  fix)  +  fff'ix)  -  ~-f'"(x)  + .  • . 

+  ---(^2,n)r^P'^-H^-)  +  9.^^), 
u'here 

^r.{x)  =  ^-^y,-/^^-"Cr)  +  E  j^  f''-Ky+t)<P2Ui)dt 

02m-l  _   1       D  f  3-  >  fl 

<.      iJ      «..U;     22--1      (2w)!-^         ^•'^'     1  0  ^  e„.(.r)  ^  1, 
and  ivhere  C  is  a  constant  as  regards  both  m  and  x,  defined  by  the  equation 

C  =  hfia)  -  ^f'{a)  +  -^f"\a) +  --(2m)r'-^'''"""^^^  "  "-^"^- 

For  the  proof  of  this  theorem  we  first  observe  that  the  conditions  for  theorem 
II,  and  hence  also  those  for  theorem  I,  are  here  fulfilled  both  for  m  =  m  and 
m  =  m  +  1 ;  also  the  conditions  that  Cm  shall  be  independent  of  m.  Upon 
applying  theorem  II  with  V.m{x)  as  given  by  (40)  and  comparing  the  result 
with  that  obtained  by  placing  m  =7/1+1  in  theorem  I,  we  obtain 

J)  [  92m  1  I  n  00 

m    (2;^  { .-W  -^s^  -  1  |/«->(x)  =  (^„^ £/-«'(.  +  ex 

Let  us  now  write  /(^m-i)  (^^-^  jj^  ^j^^  form 

-lim  (   'llf^"^Ky+t)dt 

b=co  I/O     y=^ 

and  let  Qm'ix)  represent  the  expression  Qm{x)  of  theorem  II  in  the  present  dis- 
cussion. Then,  in  case  f'--'"\x)  and /^-'"+-^(a:)  preserve  the  same  sign  it  follows 
from  (41)  that 

(42)  njix)  =  ~^^^^^  ;-^  4^m{x)f^'"^-'^  (x);         -  1  ^  ^mix)  ^  0 

and  hence,  for  the  first  expression  Qmix)  of  the  present  theorem,  we  shall  have 

0  ^  e.(.r)  ^  1 
with  which  the  first  part  of  the  theorem  becomes  established. 
1"  Cf.  Makkoff  (L  c),  pp.  131-133. 


General  Theorems  17 

If,  on  the  other  hand, /^-"'H^')  and /'^-'""'"'^  (a;)  preserve  opposite  signs  {x  >  a) 
we  shall  have  equation  (42)  in  which 

22m  1  92to— 1  1 

0  ^=  VmUV    =      92m— 1  ■'■   ~         22"i— 1 

and  thus  the  second  part  of  the  theorem  becomes  established,  upon  observing 
finally  that  we  here  have  9.„Xx)  =  0„/(a;). 

Theorem  IV.  Let  f{iv)  he  any  function  of  the  complex  variable  iv  =  x  +  iy 
ichich  is  anahjtic  throughout  all  portions  of  the  w  plane  {ic  =  co  excl.)  for  ichich 
x  ^  a.     Also,  let  it  be  supposed  that 

lim  f(x  ±  iy)e^'^-''^^  =  0;        .r  ^  a, 
where  rj  is  some  assignable  positive  quantity.     We  may  then  ivrite 
Zfix)  =  an  +    rfii-)dx  -  i/(.r)  +  ^f'ix)  -  ^f"'(x)  +  •  •  • 


where 
Qmix)  = 


i-ir  rf^'"H^-  +  edy)  -  f^'^^Kx  -  ejy) 

{2m)liJo  e'^^'y  -  1  ^    ^' 

^a;  =  1  when  m  =  0 

0  <  ^s  <  1  ichen  m  =  1,  2,  3,  •  •  •, 

and  where  Cm,  is  a  constant  as  regards  x,  defined  by  the  equation 

Crr.  =  hf{a)  -  fj'ia)  +  ^J"'{a)  -  •  •  •  +  --^2^^ ^'^~"^~'' ^'''^  ~  ^"'^"^' 

This  theorem  is,  in  fact,  a  direct  consequence  of  the  result  stated  in  §  8, 
being  obtained  from  it  by  placing  h  =  x  and  rearranging  terms. 

Generalization  of  the  Preceding  Results^^ 

10.  The  results  given  in  §  4-7  and  the  first  three  theorems  of  §  9  require 
that  the  function  f{x)  together  with  its  first  2m  derivatives  shall  be  continuous 
throughout  a  certain  specified  interval.  When  this  condition  is  not  satis- 
fied the  same  results  and  theorems  no  longer  exist,  at  least  in  general.  How- 
ever, in  cases  in  which  fix)  satisfies  the  indicated  condition  except  at  a  finite 

'*  For  a  derivation  of  the  Maclaurin  sum-formula  from  the  standpoint  of  Fourier  series,  see 
PoissoN  [1.  c).  A  still  difTcrcnt  method  may  be  found  in  Boole's  "  Treatise  on  Finite  Differ- 
ences "  (London,  1860),  pp.  80-84.  The  formula  has  been  generalized  in  various  directions  by 
Barnes;  see  Quart.  Journ.  of  Math.,  Vol.  35  (1903),  pp.  175-188;  Trans,  of  Cambridge  Philosophical 
Sac,  Vol.  19  (1904),  p.  325;  Proceedings  of  London  Math.  Soc.  (2),  Vol.  3  (1905),  pp.  253-272. 

3 


18  The  Maclaurin  Sum-Formula 

number  of  points  (at  which  discontinuity  or  uncertainty  may  exist)  we  may 
still  obtain  certain  noteworthy  results. 

Tn  order  to  show  this  we  first  observe  that  if  u  and  d  be  any  two  functions 
of  the  (real)  variable  x  which  together  with  their  first  derivatives  are  continuous 
throughout  the  interval  {a,  a -{-  h)  except  at  the  point  x  =  ^,  we  may  write 

+  )  idv  =  uv  \  -  uv  \  -  (    I        +  )  vdu 

€  being  an  arbitrarily  small  positive  quantity.     This  is,  in  fact,  a  direct  conse- 
quence of  the  ordinary  formula  for  integration  by  parts.^^ 

In  particular,  if  7/.c  be  a  function  which  together  with  its  first  2m  +  1  deriva- 
tives {u',  u",  •  •  •  «(2m+i))  jg  continuous  within  the  interval  {a,  a  +  h)  except  at 
the  point  x  =  j3  we  may  obtain  by  repeated  use  of  (43)  the  following  result 
(c-/.  (3)): 

p=l    V'  L    P=0  V-  Jz=3-a-e 

"Whence,  if  ^o,  Hi,  •  •  •  Ihm  be  the  constants  defined  by  (5)  we  may  write 
(cf.  (6)): 

2m-l  r2m-l  2m-i-/'L_      Np  -|z=3-a+e 

where 

Upon  introducing  the  function  (pim{^)  and  making  use  of  the  relations  (10) 
and  (13)  we  thus  obtain 


■L  m-1  T>    7,2fc 

hu:  =  Az..  -  \m: + E  (-  D^^^^M^'^ 


(44) 
where 


+     E  //./i^  E  ^^  l,      ^  y^ir^        +  r.(a,  A), 

+  J  jWx^''"+"^2m(.'C-«)^^. 

The  case  of  especial  interest  for  our  present  purpose  is  that  in  which  i/^  is 

taken  in  the  following  manner 

»9  As  usually  stated  (cf.  Goursat,  "  Cours  d' Analyse,"  Vol.  1  (1902),  §  85)  the  formula 
requires  that  u  and  v  with  their  first  derivatives  shall  be  continuous  throughout  the  interval  of 
integration. 


General  Theorems  19 

Ux  =    I    f(x)dx        when        a  ^  x  ^  ^  —  e, 

Wi  =  (    I         +1        )  fix)dx         ivhen        jS  +  e  ^  a^  ^  a  +  /;, 

f(x)  being  any  function  which  together  with  its  first  2m  derivatives  is  continuous 
within  the  interval  (a,  a  +  h)  except  at  the  point  x  —  /S.     Such  a  function  w^ 
together  with  its  first  2m  +  1  derivatives  will  be  continuous  except  at  a*  =  jS. 
Whence,  applying  (44),  we  may  write^° 


'"-1  iR,^2fc 

(45)  +  E  (-  I)'-' ^~ A/«'-»(a) : 

L   A=0  p=0  Pl  Jx=-f 

Let  us  suppose  lastly  that  the  interval  (a,  a  +  h)  containing  the  point  x  =  8 
is  part  of  a  larger  interval  (a,  b)  throughout  which  (except  at  a*  =  B)  f{x)  satis* 
fies  the  indicated  conditions;  also  let  us  suppose  that  a  is  one  of  the  quantities 
a,  a  -jr  h,  a  -\-  2h,  •  •  -,  b  —  h.  If  then  we  apply  formula  (16)  to/(.T)  when  con- 
sidered within  the  intervals  (a,  a),  (a  +  h,  b)  and  apply  formula  (45)  to  the 
same  function  when  considered  within  the  interval  (a,  a  +  }i)  we  obtain,  after 
adding  the  three  results  and  dividing  by  li, 

n-l  h-h  -.      /      ^3_e  r^'      \ 

Z/(a  +  g/O  =  Z/(^)  =7  +  /(.r)f/.r 

3=0  x=a  ll'   \Ja  t/g  +  e/ 

~  A  U        "^  i       )  '^^'"^ '^^^'^--^-'^  "  "^^''^ 
E  7/.^'=  Z  ^-  +  ^       ^-^  /(^+^i)(^  +  :r) 


(46) 


.where 


+        (2m -"2)T~  t/^'""''^^)  -  /^'"-'H^)]  +  /?. 


By  use  of  this  formula  instead  of  the  earlier  corresponding  one  (16)  we  arrive 

20  We  note  that/(-"(i3  +  x)  =  ?/)3+x  and  hence /(-"(/3  +  x)  ]-^=l,  =  0. 


20  The  ^NIaclaurin  Sum-Formula 

at  the  desired  theorems  corresponding  to  the  first  three  of  §  9.  Since  these  are 
long  in  statement  though  readily  supplied  we  shall  omit  them. 

Analogous  results  may  evidently  be  obtained  whcn/(.r)  presents  any  (finite) 
number  of  exceptional  points  of  the  type  just  mentioned. 

11.  Again,  the  results  stated  in  §S  and  the  fourth  theorem  of  §  9  require 
that  f{iv)  be  analytic  within  a  certain  domain.  If,  on  the  other  hand,  this 
function  presents  singularities  at  a  finite  number  of  points  within  the  domain, 
but  otherwise  satisfies  the  indicated  conditions,  we  may  readily  make  such 
alterations  as  are  necessary  to  preserve  correctness.  For  example,  let  us  sup- 
pose that  the  function  f{tc)  of  theorem  IV  satisfies  the  conditions  there  stated 
except  at  the  point  w  =  13  =  p  -\-  iq;  a  <  j)  <  x,  q  <  0.  The  theorem  will 
then  continue  to  hold  true-^  provided  that  we  subtract  from  tlie  second  member 
the  residue  r^  of  the  function 

corresponding  to  the  point  iv  —■  3.  However,  if  the  exceptional  point  occurs  at 
y;  =  fi  z=  p  -\-  iq-^  a  <  p  <  X,  q  >  0,  then  (in  view  of  the  manner  in  which  in 
§  8  the  integral  of  f(rv)  over  the  path  DEFG  was  transformed  to  one  over  the 
path  DCIIG)  the  theorem  will  continue  true  provided  we  subtract  from  the 
second  member  the  expression  r^  together  with  the  residue  r/  of  the  function 
2Trif{w)  corresponding  to  the  same  point  w  =  jS. 

Other  cases  are  those  in  which  a  singular  point  occurs  on  either  of  the  lines 
^<;  =  a  -j-  iy^  w  =  X  -\-  iy  or  at  a  real  point  iv  =  ^  <  x.  If,  in  the  last  of  these 
cases  (which  is  the  only  one  to  which  we  shall  refer  later),  the  singular  point  is 
a  pole  of  the  first  order  the  theorem  is  seen  to  continue  as  a  result  of  (29)  provided 
that  the  term 


mdx 


be  changed  to 


iim 

e=0 


(.r  ^^'i[  )-^(->^-^  -  ^^  -  ^'■^'' 

where  r^,  r/  have  the  meanings  already  given. 

Series  of  Stirling 

12.  As  a  preliminary  application  of  tlie  preceding  general  theorems  to  special 
functions  fix)  let  us  take  /(.r)  =  log  x,  a  =  amj  real  number  >  0.  We  are 
thereby  led  to  certain  well-known  results  respecting  the  series  of  Stirling. 

The  first  part  of  theorem  III  may  here  be  applied  since  we  have 


(_   1)P-1(7;  —   1)! 


"  Cf.  §  4. 


Seeies  of  Stirling  21 

Whence,  upon  observing  that 

log  xdx  =  [x  log  .r  —  .r]',  =  /»'i  +  x  log  x  —  .t;         ki  =  const. 


£ 


and  that 

x-l 

X)  log  a;  =  log  r(aO  —  log  T(a)  =  /i-2  +  log  r(.r);         7^2  =  con5^. 

x=:a 

we  obtain 

log  r(.T)  =  A  +  (X  _  ^)  log  .r  -  .r  +  j-^  ^  -  ^^  ^  +  ^  _^ 

(-ir^.-i  1 

"^  (27W  -  3)(27?i  -  2)  a;2"^-3^  ^"^^••^^' 

where  A  is  a  constant  as  regards  m  and  x  and  where 


(-  1)-+^^^       1 
(2m-  l)(2m)  a;^ 


(48)  Tmix)  =  e^(.r)  ,o^_:■,,,oI^  i^^ ;      0  ^  ©-(-^O  =  !•' 


Moreover,  by  comparing  the  above  results  with  the  well-known  formula^^ 
log  T{x)  =  I  log  27r  +  {x  —  ^)  log  x  —  x 

(49)  _   r/    1       1    i\  ,  dt 


+  Jo     Vl-e-^        t        2) 


^    t 


it  follows  (upon  placing  .x  =  00 )  that  A  =  ^  log  27r. 

Thus,  we  arrive  at  the  series  of  Stirling  (see  Preface)  and  it  appears  from 
(48)  that,  though  divergent,  the  series  may  be  used  to  compute  log  T{x)  with  but 
slight  error  when  x  (real  and  positive)  is  large.  In  fact,  the  first  term  neglected 
is  seen  to  constitute  an  upper  limit  to  the  error  committed  by  breaking  off  the 
series  at  any  one  point.  This  fact  was  pointed  out  by  CArciiY-^  in  1843  through 
an  independent  investigation  based  upon  formula  (49),  he  also  noting  in  this 
connection  the  possible  value  of  divergent  series  in  computation.  Cauciiy's 
work  was,  however,  confined  to  this  one  series  and  in  this  it  appears  that  his 
results  might  have  been  obtained  much  more  directly,  as  indicated  in  §  12, 
from  the  earlier  general  investigations  of  PoissoN  and  Jacobi  relative  to  the 
Maclaurin  Sum-Formula. 

We  add  that  the  value  of  the  constant  K  may  be  obtained  independently  of 
formula  (49)  by  use  of  the  well-known  formula  of  Wallis  expressing  the  value 
of  x/2.25 

22  In  the  present  case  it  may  be  shown  that  0  <  0„,(x)  <  1.     See  Malmsten  {I.  c),  p.  75. 

2'  Usually  attributed  to  Binet. 

2''  See  Comples  Rendus  de  I' Acad,  dcs  Sciciices,  Vol.  17  (1S43),  pp.  370-376. 

2*  See  Markoff  [l.  c),  p.  134. 


22  The  IMaclauein  Sum-Formula 

Preliminary  Discussion  of  Asymptotic  Series 

13.  The  formula  of  Stirling,  by  means  of  which  the  function 

log  r(.r)  -  (x  —  ^)  log  x+  X 

may  be  identified  with  a  certain  divergent  power  series  in  l/x,  affords  an  illustra- 
tion of  an  important  class  of  developments  known  as  asymptotic  series.  We 
proceed  to  give  at  this  point  a  brief  exposition  of  the  general  features  of  this 
subject,  leaving  its  further  development  and  applications  for  later  chapters, 
especially  chapters  II  and  III. 

Following  PoiNCARE,  we  adopt  the  following  definition:-^ 

"  A  power  series  of  the  form 

(50)  oo  +  Qi  (".)+  fl2  (")  +  ••• ;        flo,  «!,  ^2,  •  ■ '  constants 

is  said  to  represent  asymptotically  the  function  f{x)  for  large  positive  values  of  x 
whenever 

lim  X-  [fix)  -  («o  +  a,fx  +  a^fx'  +  •  •  •  +  aj.r")]  =  0; 

(51)  ^=+'» 

n=  0,  1,  2,  3,  •••."" 

Thus,  for  a  given  value  of  n  the  difference  between  the  function  and  the  sum 
of  the  first  n  +  1  terms  of  its  corresponding  asymptotic  series  (in  case  one  exists) 
vanishes  to  a  higher  order  than  the  nth  when  x  =  +  oo ,  as  would  be  the  case  in 
particular  if  the  series  were  convergent.  Symbolically,  the  above  relation  is 
expressed  as  follows: 

(52)  fix)  -  Oo  +  ajx  +  a,lx'  +  •  •  • . 

Several  general  observations  are  here  desirable.  First,  a  given  function  fix) 
can  be  represented  asymptotically  in  but  one  way.     In  fact,  we  have  from  (51) 

(53)  fix)  =  ao+  ailx  +  a.lx'  +  •  •  •  +  an-xlx--'  +  ""^'"^""^ ;        lim  €„(a:)  =  0 

26  See  Ada  Math.,  Vol.  8  (18S6),  p.  29G. 

2'  In  this  definition  no  restrictions  are  placed  upon  (50)  as  regards  convergence  or  divergence. 
However,  in  the  usual  applications  the  series  is  divergent  for  all  values  (positive)  of  x,  but  as  an 
instance  in  which  the  contrary  is  the  case  we  have 

X         3?         X^ 

In  the  most  important  applications  (cf.  Chapters  II  and  III)  /(x)  is  a  function  (either  given 
explicitly  or  else  determined  implicitly  as  a  solution  of  a  linear  differential  or  difference  equation) 
capable  of  analytic  continuation  into  the  complex  field,  being  in  fact  analytic  throughout  the 
finite  plane  with  the  exception  of  points  (finite  or  infinite  in  number)  situated  upon  a  finite  number 
of  straight  fines  radiating  from  the  origin  and  having  the  point  x  =  «=  as  a  non-polar  singularity. 
For  further  criticisms  upon  the  definition  of  asj'mptotic  series  sec  Thom6,  Journ.  fiir  Math., 
Vol.  24  (1904),  pp.  152-156;  Van  Vleck,  The  Boston  Colloquium  Lectures  (New  York,  Mac- 
millan,  1905),  pp.  77-85;  Watson,  Philosophical  Trans.,  Vol.  211A  (1911),  pp.  279-313. 


Asy:mptotic  Series  23 

and  in  case  we  had  also 
fix)  =  60  +  br/x  +  b,/x^'  +  •  •  •  +  6n-i/a:"-^  +  ^"  ^^^'^'^ '  ^'"^  ^"'^^)  =  ^ 

•C  a;z=-|-oo 

we  should  have 

(ao  -  bo)  +  (ai  -  fei)  ^  +  (a2  -  62)  -^  +  •  •  •  +  («n-i  -  6n-i)  -;ii:i 

.an-bn-\r  enjx)  —  €„'(a:)  ^ 

Whence,  Oo  =  &o,  as  results  from  the  last  equation  by  placing  x  =  +  x> .  Making 
use  of  this  relation  in  (54),  multiplying  both  members  by  x  and  proceeding  as 
before,  we  obtain  ai  =61,  •  •  •,  etc.  The  converse  of  the  above  statement  is, 
however,  not  true  as  appears  directly  when  we  note  that  if  f(x)  is  represented 
asymptotically  by  (50)  so  also  is,  for  example,  the  function  f(x)  +  e~^.^^ 

Again,  it  is  desirable  for  the  sake  of  clearness  to  note  that  asymptotic  series 
in  general  cannot  be  used  for  purposes  of  computation  in  the  sense  in  which 
Stirling's  series  can  be  used  to  compute  log  r(.T).  In  fact,  no  information  is 
at  hand  respecting  the  error  committed  by  stopping  at  any  preassigned  term.-^ 
There  are,  however,  numerous  and  important  asymptotic  developments^''  which, 
like  the  series  of  Stirling,  are  derivable  by  use  of  the  Maclaurin  Sum-Formula 
and  for  such  the  limit  of  error  may  usually  be  fixed  by  means  of  the  formulas 
then  present  for  the  remainder.  But  in  all  cases,  the  asymptotic  development 
furnishes  information  as  to  the  behavior  of  the  function  when  x  is  very  large. 
Thus,  the  expressions 

ao,  Go  +  ai/x,  Go  +  ai/x  +  ai/x^,  •  •  •,  ao  +  ai/x  +  ai/x'^  +  •  •  •  +  cim/x"' 

constitute  a  series  of  successive  approximations  to  the  value  of  /(.r)  provided 
that  X  is  sufficiently  large.     Furthermore,  we  have 

lim/(a:)  =  Oo 

z=+co 

(55)  lim^a;[/(.r)  -  Oo]  =  cti 

lim  x^'ifix)  —  ao—  ai/x  —  ailx^  —  . ..  —  a„_i/.r"-i]  =  o„. 

Conversely,  when  the  behavior  of  f{x)  for  large  positive  values  of  x  is  known, 
the  equations  (55)  serve  to  determine  the  coefficients  ao,  ai,  02,  •  •  •  of  the  corre- 
sponding asymptotic  development  if  one  exists. 

28  By  adopting  a  more  limited  definition  of  asymptotic  scries  than  that  of  PoiNCARfi,  Watson 
has  obtained  a  noteworthy  theorem  upon  this  question  of  uniqueness.  See  Philosophical  Trmis., 
Vol.  211A  (1911),  p.  300. 

29  For  noteworthy  exceptional  cases,  see  Stieltjes,  Annales  de  I'Ecole  Normale,  ^'ol.  13  (18SG), 
pp.  201-202. 

3"  This  is  true  in  general  of  the  developments  considered  in  Chapter  II. 


24 


The  Maclaurix  Sum-Formula 


14.  The  following  consequences  of  the  definition  (51)  are  especially  note- 
worthy:^^ 


7/ 


then 
(a) 

ib) 


fix)  ~  oo  +  fli/.'K  +  aa/.v"  +  •  •  • , 
(p{x)  ~  6o  +  bi/x  +  bojx-  +  •  •  • 

Oi  ±  6l    ,     02  =t  &2    , 

fix)  ±  <pix)  -  (ao  ±  6o)  +  -^7—  + ^  + 


X  X- 

fix)  •  (fix)  '^  eo  +  Ci/.r  4-  Co/a:-  +  •  •  • , 
where  c„  =  oo^n  +  aibn-i  +  a2&n-2  +  •  •  •  +  Cnho', 
(c)  /(a^)/^(ar)  ~  cfo  +  (/i/.t  +  d^/x'  +  •  •  • , 

provided  that  bo  +  0,  f/ze  coefficients  do,  di,  do,  •  •  •  ^emgr  determined  hy  the  equations 
'  cfo  =  b^d^ 
a\  =  6if?o  +  ^of/i 
On  =  bndii  4-  6„_iJi  +  •  •  •  +  6orfr,; 


{d) 


provided  that  Oo  =  ai  =  0." 

In  other  words,  asymptotic  series  are  subject  to  the  same  laws  of  addition, 
subtraction,  multiplication,  division  and  term  by  term  integration  as  convergent 
power  series  in  l/x. 

For  the  proof  of  (a)  we  have  but  to  note  that 

an+  enix) 


lim  e„(.r)  =  0, 


bn  -\-   en'ix) 


fix)  =  ao-\-  ajx  +  a2/a;2  +  •  •  •  +  a„-i/^"  ^  + 

ipix)  =  6o  +  bilx  +  h.Jx'  +  •  ■  •  +  bn-i!x--'  + 

Thus,  we  may  write 

xr  ^    ,       /^        r      ^l  \    I              y     f          _i_7       ^^J_  («n  rb  bn)  +  r7n(a-) 
/(a-)  ±  ifix)  =  (rto  4=  feo)  + h  (on-i  ±  On-i)  ^:^„_i  +  -^ 


lim  e/(.r)  =  0. 

X=-\-  00 


lim  r?„(.r)  =  0. 


As  regards  (6),  let  us  indicate  by  *S„(a-),   r„(a:)  and  2„(.r)  respectively  the 
sums  of  the  first  n-\-  I  terms  of  the  three  series  in  question.     Placing  for  brevity 

fix)   =  /,     <pix)    =    ^,     €nix)    =    6,     6„'(.r)    =    €',     Sn(.r)    =    S,      T  nix)    =    T,     2„(.t)   =    S, 

lim   =  lim,  we  shall  have 


3'  See  PoixcARf  (L  c),  pp.  297-301. 


Asymptotic  Series  25 

(56)  /=S  +  :^,         ^=r  +  f^;        Iim6  =  lim6'  =  0 

and 

where  P  is  a  polynomial  in  x  of  degree  no  higher  than  the  {n  —  l)st. 
Whence, 

or 

x^lf  -  <p  -  i:]  =  fe'  +  cpe+  {P  -  ee')^-. 

Now,  lim/  =  cfo;  lini  cp  =  bo  from  which  it  follows  that 

lima;"[/-  (^  -  S]  =  0. 

For  the  proof  of  (c)  let  us  use  the  same  notation  as  above  except  that  2  shall 
represent  the  sum  of  the  first  n  -\-  1  terms  of  the  series  in  which  the  coefficients 
do,  di,  c?2,  •  •  •  appear.     Then,  using  equations  (56)  we  shall  have 

_^   , 

and  since  lim  S  =  ao,  hm  T  =  6o  =1=  0  it  follows  that  hm  ^"tj  =  0. 
Moreover, 

y^=  2  +  w;        hm.T"a;  =  0. 

Whencs, 


Hm  .T"  f  -  -  2  j  =  Hm  x^'irj  +  co)  =  0. 


The  proof  of  (d)  is  readily  supplied.     We  have  from  (53)  when  ao  =  ai  =  0 


/ 


S{x)dx  -  ^-t  2^^.2  -t-  3.^3  i-  \-  (,^  _  2)a;"-2  "^  (n  -  l)a:~-i ' 


/„(a;)  =  X"  '  J      ^^dx 


and,  since  lim  e„(a;)  =  0,  we  may  say  that  corresponding  to  an  arbitrarily  small 
positive  quantity  5  there  exists  a  constant  x^  such  that  |e„(.r)|<  5;  x  >  x^. 
Whence, 

,  .     M  .r.      rdx  d 

M.r)|^.x--5j^    -.=  --,;         x>x, 
so  that  lim  r]n{x)  =  0. 


26  The  Maclaukin  Sum-Formula 

In  distinction,  however,  to  the  properties  of  convergent  power  series,  the 
term  by  term  derivative  of  the  asymptotic  development  of  f(x)  will  not  neces- 
sarily be  the  asymptotic  development  of /'(a-).  This  is  most  easily  shown  by  an 
example.     Thus, 

(57)  /(:r)  =  e"-  sin  (e^)  ~  0  +  "  +  ^+  •  * ''' 

but  since /'(.^)  =  —  e"""  sin  {e"")  +  cos  (e""),  the  expression  hm/'(.r)  is  oscillatory 
so  that  not  only  does  the  term  by  term  derivative  of  the  series  (57)  fail  to  repre- 
sent/'(a-)  asymptotically,  but/'(.T)  permits  of  no  such  representation  whatever. 
However,  if 

fix)  ~  «o  +  -  +  ;^  +  •  •  • 

and  if /'(a)  is  known  to  be  developable  asymptotically,  then 

„,,  ,  ai       2a2       3cf3 

(58)  •^^^■^'^-.^-'^-"^ • 

In  fact,  if /'(.r)  were  developable  asymptotically  in  any  other  way  than  (58) 
it  would  follow  from  (d)  of  the  above  results  that  /(.r)  was  developabl ;  asymp- 
totically in  two  different  w^ays. 

15.  In  addition  to  the  properties  (a),  (b),  (c)  and  (d)  of  §  14  we  note  also  the 
following  general  result: 

''Let 

fix)  =  ao  +  tvix) ;         ivix)  ~  ^  +  J  +  * ' ' 

and  let  Fif)  he  a  function  of  x  through  f  which,  tvhen  written  in  the  form  Fiao  +  w), 
is  developable  as  folloics: 

Fiao  +  w)  =  Fiao)  +  F'iao)w  +  ^^  w' -\-  ■  - - 

^""^^  .F'^'-^'M     „_,    ,    F(")(ao)  -f-  6n(2tO     „ 

(n  —  1)!  nl  u,=o 

ias  happens  in  particular  when  Fiao  +  ?y)  is  analytic  at  xo  —  0).     Then  we  may 
write 

f(/)~f(«.)+'>|+---+f-:+---, 

where  p\,  jh,  •  •  • ,  Pn  are  the  coefficients  of  the  successive  powers  of  l/x  ohtai?ied  by 

substituting  into  (59)  (  exclusive  of  the  term  "       w"  1  the  first  n  terms  of  the  given 

asymptotic  development  of  wix)." 

32  Cf.  BuoMwicii,  "  Infinite  Series  "  (London,  1908),  p.  334. 


Asymptotic  Series  27 

In  fact,  from  (b)  of  §  14  we  may  write 

F{ao)  +  F'{ao)w  -| 2]^^'  +  •  • '  H ^^l —  '^  F{ao)  +  —  +  -:$+  •  •  •, 

and  hence  (59)  may  be  written  in  the  form 

Fiao  +  w)  =  F(ao)  +  -  +  1^+  •  •  •  H ^n ^  'T^^  '        ^1°^  '^nOx')  =  0. 

If  we  now  write  €n{w)io'^  in  the  form 

and  observe  that 

lim   en{w)w'^x'^  =■  0 

a;=oo 

we  obtain  the  desired  result. 

16.  We  note  in  connection  with  the  definition  (51)  that  we  have  supposed  x 
real  and  positive.  More  generally,  f{x)  is  said  to  be  represented  asymptotically 
by  the  series  (50)  throughout  an  infinite  region  T  (usually  a  sector  with  center 
at  a:  =  0)  of  the  complex  plane  when,  for  all  corresponding  x  values,  the  equation 
(51)  exists  in  which  lim  is  substituted  for  lim  .     In  the  case  frequently  pre- 

\x  I  =00  a-=  +  «) 

sented  of  a  single-valued  function  f{x)  having  an  essential  singularity  at  the 
point  X  =  00 ,  we  note  that  the  above  mentioned  region  cannot  completely  sur- 
round the  point  a;  =  00 ,  since  we  should  then  have  lim  f{x)  =  ao  for  all  methods 

|x|=oo 

of  increase  of  \x\,  thus  contradicting  the  hypothesis  that  the  point  a;  =  00  is 
essentially  singular. 

Again,  if  f{x)  and  the  region  T  be  given,  we  observe  that  the  necessary  and 
sufficient  condition  that  f{x)  be  developable  asymptotically  throughout  T  is 
that  there  exist  a  set  of  constants  oo,  ai,  CI2,  •  •  •,  On,  •  •  •  satisfjdng  relations  (55), 
it  being  understood  that  the  values  of  x  appearing  in  these  relations  are  confined 
to  T.  In  fact,  if  (55)  exist  we  have  (51)  and  conversely.  The  same  relations 
(55),  when  employed  as  a  sufficient  test  for  the  existence  of  an  asymptotic  de- 
velopment for  f{x)  throughout  T,  are  usually  difficult  to  apply  and  hence  of 
little  value  in  practice,  since  f{x)  is  not  in  general  so  given  that  it  is  possible 
to  determine  whether  the  indicated  limits  (representing  ao,  ai,  a2,  •••)  exist. 
A  sufficient  test  which  has  a  wider  field  of  applicability  is  supplied  by  the  fol- 
lowing 

Theorem  V.'^^    Let  fix)  he  a  function  of  the  complex  variable  x  analytic  within 

and  upon  the  boundary  of  a  certain  infinite  region  T  of  the  x  plane,  the  point  .r  =  00 , 

hoivever,  being  excluded.     Also,  let  (p{x)  =  f{l/x)  and  let  T'  be  the  region  {having 

33  Cf.  Ford,  Bulletin  Soc.  Math,  de  France,  Vol.  39  (1911),  p.  348.  Line  13  should  here  read 
"  le  point  a;  =  00  toutefois  etant  exclu." 


28  The  IMaclaurin  Sum-Formula 

the  point  .r  =  0  u'pon  its  boundary)  obtained  from  T  by  means  of  the  transformation 
X  =  l/.r'.     //,  then,  for  values  of  x  in  T'  the  foil oiving  limits  exist: 

lim  ^(.r),         lim  ^'(x-),         lim  ^"(.r),  •••,         lim  ^^"^(a;), 

and  are  represented  respectively  by  (p{0),  ^'(0),  •  •  •,  (p''''\0),  •  •  -  {these  values  being 
assumed  independent  of  the  direction  of  approach  of  x  to  0  in  T')  we  may  write  for 
values  of  x  in  T 

fix)  -  flo  +  ai(l/.r)  +  aoil/xY-  +  •  •  •  +  a.(lAr)-  +  •  •  • 
where 

cik  =      ^.,      ;        /t  =  0,  1,  2,  3,  •  •  •,  ?i,  •  •  • . 

In  order  to  prove  this  Theorem  we  shall  begin  by  establishing  the  following 
Lemma  in  the  general  theory  of  functions: 

Lemma  I.  "  Let  (p{x)  be  a  function  of  the  complex  variable  x  analytic  within 
and  upon  the  boundary  of  a  certain  region  T'  of  the  a:-plane,  exception  being 
made,  however,  of  the  point  x  =  0  situated  upon  the  boundary  at  which  point 
<p{x)  may  have  any  character.  If,  then,  for  values  of  x  within  T'  the  following 
n  -{-  2  limits  exist: 

Hm  (fix),        Hm  (p'(x),        hm  (p"{x),         ••-,         Hm  ^^«+i)(a:) 

a:=0  x—Q  z=0  x=0 

and  are  represented  respectively  by  ^(0),  (p'{0),  <p"iO),  •  •  • ,  ^^"+^^(0)  (these  values 
being  assumed  independent  of  the  direction  of  approach  of  a*  to  0  in  T')  we  may 
write  for  values  of  x  in  T' 

(p{x)  =  «o  +  aix  +  aox-  +  •  •  •  +  an-i.r"~^  +  [n„  +  r„(x)].t";         lim  r„(.r)  =  0, 


x=0 


ao,  c'l,  ao,  •  •  • ,  an  being  constants  determined  by  the  equation 

(P^''HO) 
a,  =  ^j^\k=^0,l,2,  •••,n)." 


In  fact,  under  the  above  hypotheses  we  may  write  for  any  value  of  x  in  T' 
(60) 


<p(x)  =  <p(c)  +  <p'{c){x  -c)  +  ^(x-c)'-+  '■■+  ^-^  (x  -  c)" 


+ 


^J\x  -  t)-cp^-+'Kt)dt, 


where  c  represents  a  fixed  value  of  x  taken  within  T'  and  arbitrarily  near  to  0 
and  where  (at  least  if  x  and  c  are  each  taken  sufficiently  near  to  0)  the  inte- 
gration is  understood  to  take  place  along  the  straight  line  joining  the  point  x  =  c 
to  the  point  x  =  0.  The  existence  of  (60)  may  be  readily  verified  by  performing 
an  integration  by  parts  n  times  upon  the  last  term  in  the  second  member.^^ 
3^  Cf.  GouRSAT,  "  Cours  d'iVnalyse,"  Vol.  1  (1902),  §  86. 


Asymptotic  Series  29 

If  in  (GO)  we  now  allow  c  to  approach  the  limit  zero  through  values  that  lie 
within  T'  {x  fixed),  and  if  we  introduce  at  the  same  time  our  hypotheses  con- 
cerning the  existence  and  meaning  of  ^(0),  ^'(0),  ^"(0),  •  •  •,  (;c>^"^(0),  we  obtain 


<^"(0)    ,   ,  ,   <^("-"(0) 


,.n-l 


where 

(62)  r„(.T)  =  [{^—^y  cp'^+'Kt)dt. 

In  order  to  complete  the  proof  of  the  Lemma  it  thus  remains  but  to  show 
that  with  r„(.T)  defined  as  in  (62)  we  shall  have  lim  r„(.T)  =  0  provided  always 

that  X  remain  in  T'. 

Now,  for  all  values  of  t  on  the  line  of  integration  in  (60)  we  have 


X  —  t 

X 


1. 


Moreover,  it  follows  from  our  hypotheses  that  we  may  find  a  positive  constant  M 
(independent  of  x)  such  that  for  all  values  of  x  in  T'  we  may  write  |  ^'^"+^^  (.r)  |  <  M. 
Whence,  if  we  place  \x\=  p  we  shall  have  for  the  given  value  of  x 


|r„(.T)|<  M  jdp  = 


Mp 


from  which  the  desired  result  becomes  evident. 

Theorem  I  follows  as  an  immediate  consequence  of  the  Lemma  upon  sub- 
jecting the  function /(.t)  and  the  region  T  mentioned  in  the  theorem  to  the  trans- 
formation X  =  1/x'. 

We  note  also  that  if,  instead  of  having  /(.r)  defined  throughout  a  complex 
region  T,  it  is  given  as  a  function  of  a  real  variable  x  within  the  infinite  interval 
(a,  +  °o ),  we  may  obtain  in  like  manner  the  following  Lemma  and  corresponding 
Theorem : 

Lemma  II.  "  Let  (p{x)  be  a  function  of  the  real  variable  x  which,  together 
with  its  first  n  -\-  1  derivatives,  is  continuous  within  the  interval  (0,  h),  the  end 
point  X  =  0  being  excluded.  If,  then,  the  limits  ^(+  0),  ^'(+  0),  (^"(+  0), 
•  •  •,  (p("+^^(+  0)  exist,  we  may  write  for  values  of  x  in  (0,  6) 

<p{x)  =  ao  +  (iix  +  atx"^  +  •  •  •  +  fln-i-^'""^  +  Wn  +  '-uGiOl-i'";        lim  r„(.r)  =  0, 


a=+0 


ao,  a\,  02,  ••-,«„  being  constants  determined  by  the  equation 
ak  =  ^^'^^1  ^^  {k  =  0,  1,  2,  ■■',11):' 


30  The  Maclauein  Sum-Formula 

Theorem  VI .  Let  f(x)  be  a  function  of  the  real  variable  x  which,  together 
with  its  derivatives  of  all  orders  is  continuous  throughout  the  infinite  interval  (a,  +  <» ) . 
If  upon  placing  ip{x)  =  fiXJx)  the  following  limits  exist: 

<^(+0),        <p'(+0),        ^"(+0),         •••,        .p(">(+0), 

we  may  write  for  values  of  x  in  {a,  -\r  oo ) 

/(.)~a„+a.(l)+«.(y'+---+a,(^^)"+..., 

where 

<P''\+  0)  z       n   1    0   7 

17.  We  observe  finally  that  the  use  of  the  symbol  ~  is  frequently  broadened 
as  follows:  "  If/,  ^  and  \p  are  three  functions  of  x  such  that  in  the  sense  of  §  13 
we  have 

1^  x       x^ 

the  same  relation  may  be  written  in  the  form 

(63)  /~co+aoi/'  +  —  +  -^+  •••. 

Thus  we  write  when  x  is  real  and  positive  (cf.  §  12) 

log  r(a-)  ~  (a:  -  I)  log  a:  +  a;  +  I  log  27r  +  ^  -  -  ^^  -3+  •  •  •. 

Relation  (52)  may  furthermore  be  written  in  the  simple  form  /  ~  ao,  this 
being  especially  true  in  applications  of  the  theory  (such  as  the  determination 
of  lim  /(.r))  wherein  the  values  of  the  coefficients  Oi,  02,  03,  •  •  •  play  no  part. 

Likewise,  relations  of  the  form  (63)  may  be  written  /  ~  ^  +  oo^- 


CHAPTER  II 

THE  DETERMINATION  OF  THE  ASYMPTOTIC  DEVELOPMENTS  OF  A  GIVEN 

FUNCTION 

18.  Let  F(z)  be  a  given  function  of  the  complex  variable  z  defined  throughout 
the  finite  z-plane  and  such  that  (a)  the  point  z  =  oo  is  a  non-polar  singular  point 
and  (b)  when  1 2 1  is  sufficiently  large  and  arg  z  lies  within  a  given  sector  A  (center 
at  z  =  0)  there  exist  two  functions  /^(z)  and  (p),{z)  each  defined  throughout  A 
and  a  set  of  constants  ao,  a>  Qi.  aj  (I2,  \,  ' ' '  dn,  k,  '  ' '  such  that  for  values  of  2  in  A 
w^e  have 

r/    N           ^  /    ^     I            /    N   r                ,     ai.  A     ,     ^2,  A     ,                   ,     ttn,  A  +  ^A.nCg)  1 
F(Z)  =/a(2)  +  <^a(2)[«0.A+-^H-^H 1 -n J; 

lim  WA.nCz)  =  0. 

|2|=00 

Then,  according  to  the  definition  of  §  13  and  the  remarks  of  §  17,  we  may  write 
for  the  indicated  values  of  z 

F(z)  ~ A(z)  +  <p,iz)  [ao.x  +  ^'-^^^'+  '-']' 

This  form  of  asymptotic  development  is  of  frequent  occurrence  and  of  prime 
importance  in  analysis.  The  problem  of  determining  for  a  given  Fiz)  and  A 
the  corresponding /^(z),  <px{z)  and  ao,x,  «i.a>  «2.a,  •  •  •  (assuming  that  they  exist) 
is  usually  one  of  considerable  difficulty  and,  when  regarded  in  a  general  sense, 
is  one  for  which  but  fragmentary  results  exist  at  the  present  time.  The  known 
determinations  appear  to  be  either  those  for  special  functions  of  importance  in 
Mathematical  Physics,  such  as  Bessel's  function  J„(2)/  or  for  certain  types  ot 
integral  functions,  notably  those  defined  by  infinite  products." 

In  the  present  Chapter  it  is  proposed  to  show  how  the  general  theorems  of 
Chapter  I  may  be  used,  at  least  in  certain  cases,  to  make  the  above  indicated 
determinations.  In  doing  this  we  shall  merely  consider  certain  special  functions 
F(z).  No  attempt  will  be  made  to  obtain  theorems  of  great  generality,  partly 
because  of  the  difficulty  of  such  an  undertaking,  but  chiefly  because  of  the  bcliet 
that  a  few  well-chosen  illustrations  suffice  to  adequately  impart  the  spirit  and 
possibilities  ot  the  method  employed.     In  each  of  the  functions  F(x)  considered, 

'See  for  example  Lommel,  "Studien  iiber  die  Bcsscl'schcn  Functionen  "  (1868),  §  17. 

'^  See  for  exumplc  Barnes,  Philosophical  Transactions,  Vol.  199A  (1902),  pp.  411-500; 
ibid.,  Vol.  206A  (1906),  pp.  249-297.  Each  of  these  memoks  contains  an  extended  bibliography 
of  the  subject.  See  also  Mattson,  "Contributions  h  la  Th^orie  des  Fonctions  entiSres  "  (Thdse), 
Upsala,  1905. 

31 


32  Determination  of  Asymptotic  Developments 

only  the  functions /a (2),  ^^(2)  and  the  first  one  of  the  constants  a„,x  which  is 
not  equal  to  zero  are  determined,  since  these  three  determinations  constitute 
what  is  essential  to  the  study  of  the  behavior  of  F{z)  for  large  values  of  |2;|. 
The  method,  however,  permits  equally  of  the  determination  of  any  one  of  the 
coefficients  fl„,  a- 

The  functions  F{z)  considered  fall  into  two  classes:  (a)  those  defined  by 
infinite  products  and  {b)  those  defined  by  infinite  series.  Under  (a)  we  have 
eventually  considered  (§§  24-28)  the  asymptotic  behavior  of  the  general  integral 
function  of  order  >  0 — a  problem  to  which  considerable  attention  has  been 
devoted  in  recent  years^  and  in  connection  with  which  we  have  entered  into 
considerable  detail  owing  to  the  importance  of  this  and  other  analogous  con- 
siderations in  the  general  theory  of  functions.  Under  (6)  we  have  eventually 
considered  (§§28,  29)  the  asymptotic  behavior  of  functions  defined  by  power 
(]\Iaclaurin)  series — a  subject  of  evident  importance  owing  to  the  essential 
role  of  such  series  in  anahsis.  The  treatment  for  the  latter  is  brief  and  indeed 
but  fragmentary,  yet  it  is  believed  that  the  most  important  known  results  (aside 
from  those  which  concern  the  solutions  of  linear  differential  or  linear  difference 
equations)^  have  been  indicated. 

The  determination  of  the  asymptotic  character  of  functions  defined  in  other 
ways  than  as  infinite  products  or  infinite  series  might  well  have  been  considered 
also  in  the  present  chapter,  as  likewise  the  corresponding  problem  for  certain 
noteworthy  special  functions.^  We  have,  however,  limited  ourselves  in  the 
manner  indicated  above,  feeling  that  not  all  aspects  of  the  subject  could  receive 
treatment  within  the  limits  of  the  chapter  while  those  of  the  greatest  permanence 
in  the  general  theory  of  functions  have  been  included,  we  believe,  through  the 
present  selection. 

19.  Example  1.     To  obtain  asymptotic  developments  for  the  function 

°°  1 

^^^  ^^'^=S(2M=T)H^^' 

We  here  choose  a  function  which,  as  a  result  of  the  well-known  formula^ 
tan  2;       Y^  1 

^'         "="(2^+1)2^-22 

*  See  note  at  the  bottom  of  page  44. 

*  See  Chapter  III. 

*  For  miscellaneous  investigations  of  this  description,  see  Barnes,  Edinburgh  Trans., 
Vol.  19  (1904),  pp.  426-439;  Proceedings  London  Math.  Soc,  Vol.  3  (1905),  pp.  273-295;  ibid., 
Vol.  5  (1907),  pp.  59-116;  Transactions  Cambridge  Philosophical  Soc,  Vol.  20  (1907),  pp.  253- 
279;  Quarterly  Journ.  of  Math.,  Vol.  38  (1907),  pp.  116-140;  Hardy,  Quarterly  Journ.  of  Math., 
Vol.  37  (1906),  pp.  369-378;  Littlewood,  Transactions  Cambridge  Philosophical  Soc,  Vol.  20 
(1907),  pp.  323-370. 

^  See,  for  example,  Tannery's  "  Introduction  h  la  Thdorie  des  fonctions  d'une  variable  " 
(Paris,  1886),  §  117. 


Special  Functions  33 

may  be  evaluated  in  the  form 

TT  e''^  —  1 

(2)  n^-i.7^x- 

and  this  fact  will  enable  us  to  check  our  subsequent  results. 

In  order  to  obtain  the  asymptotic  developments  of  F{z)  as  defined  by  (1), 
let  us  place 

^'^'^^  ^  {2io  +  1)2  +  z" 

and  regard  z  as  having  any  fixed  value  z  —  j)  -]r  iq,  i  =  V—  1,  lying  in  a  sector 
(center  at  z  =  0)  situated  in  the  right  half  of  the  z  complex  plane  and  having 
neither  of  its  bounding  lines  coincident  with  the  axis  of  pure  imaginaries.  Then 
fz{w),  considered  as  a  function  of  the  complex  variable  w  =  x  +  iy,  satisfies  the 
conditions  demanded  by  Theorem  IV  {a  =  0)  of  Chapter  I,  except  that  in  case 
I  g  I  >  1  the  same  function  will  present  a  single  pole  of  the  first  order  at  the  right 
of  the  pure  imaginary  axis,  this  pole  being  situated  at  the  point  iv  =  l{—  1  —  iz) 
if  g  >  1  and  at  the  point  iv  =  |(—  1  +  iz)  if  g  <  —  1. 

Thus  we  may  apply  the  theorem,  subject  to  the  remarks  of  §  11,  in  order  to 
obtain  an  expression  for  the  sum 

x-l 

H  fz(x);         x>p. 

x=0 

We  shall  now  distinguish  between  the  following  four  cases:  (a)  |g|<  1, 
(b)  q>l,{c)q<  -lAd)q=  ±  1. 

In  (ft)  we  may  make  direct  application  of  the  theorem.  Taking  m  =  0,  we 
thus  obtain 

^^^     £  {2x  +  1)2  +  z^  ^^'""^X    (2a:  +  1)^  +  z^  ~  ^  '  {2x  +  1)^  +  z^  +  "^^•''^' 

where 

(4)  U^{x)  =  -  *  J      g2^y  _  I — ■ dy 

and 

(5)  c.  =  2(rT^)  ~  "^^^^• 

In  these  results  let  us  now  allow  x  to  increase  indefinitely,  observing  that 

r            dx                  If               2a:  +  1 1^="       tt         1  1 

f     /o     I    i\2   I — 2  =  TT    ^rc  tan =  -. —  arc  tan  - 

Jo     (2.r  +  If  -\-  z^      2z\_  z       J^=o       43       2z  z 

and  that  lim  Q.z{x)  =  0.     We  obtain 

a;=<» 

4 


34  Determination  of  Asymptotic  Developments 

r^.   .  -^     ,  1  1  1 

4z      2(1  +  Z-)       2z  z 

(6) 

p  r  1 1 1       dy 

+  Vo     L  (1  +  2iyr-  +z'       (1  -  2i»2  +  2^  J  e'^y  -  1  * 

Upon  developing  the  various  terms  of  the  second  member  in  ascending  powers 
of  1/z,  we  thus  reach  (Theorem  V,  Chapter  I)  the  relation 

in  which  the  coefficients  02,  oa,  cig,  •  •  ■  may  be  evaluated  to  any  desired  point.'' 
In  case  (6),  equation  (3)  and  hence  (6)  also,  will  continue  to  hold  true  ac- 
cording to  §  11  provided  that  we  subtract  from  its  second  member  the  residue 
of  the  function 

/o^  M 

""^^  [{2w  +  1)'  +  zV'""  -  1] 

at  the  point  w  =  —  ^{l  -\-  iz)  which  (residue)  is  readily  found  (cf.  (30),  Chapter 
I)  to  be  7r/22(e"^  +  !)•  Since,  for  values  of  z  within  the  proposed  sector,  this 
function  is  developable  asymptotically  in  the  form  (50)  of  Chapter  I  with 

flo  =  «i  =02=  •  •  •  =0, 

it  follows  that  relation  (7)  holds  true  also  in  case  (6). 

Similarly  in  case  (c)  we  have  equation  (6)  except  (cf.  §  11)  that  we  must  now 
subtract  from  its  second  member  the  residue  of  (8)  at  iv  =  ^{1  —  iz)  and  also 
that  of  the  function  2Trifz{iv)  at  the  same  point;  i.  e.,  we  must  subtract  the  ex- 
pression 


2z(e-'''  +1)   '   2z      2z(e'"'  +  1) " 

Thus,  as  in  case  (c)  we  see  that  relation  (7)  again  holds  true. 

Moreover,  the  same  relation  continues  in  case  (d)  as  appears  by  writing 
F{z)  in  the  form 

1  CO  1 


1  +  22  '  „^  (2r^  +  1)2  +  22 

and  applying  the  method  of  case  (a)  to  the  summation  here  appearing;  also 

recalling  that  in  one  and  the  same  region  there  can  exist  but  one  asymptotic 

development  for  a  given  function. 

Similarly,  if  we  note  the  effect  in  (4)  of  supposing  the  real  part  of  z  to  be 

negative,  we  find  that  when  z  is  situated  in  a  sector  lying  within  the  left  half  of 

^  It  may  be  noted  that  by  using  a  sufficiently  large  value  of  m  in  applying  Theorem  IV  (Chap. 
I)  we  may  obtain  any  one  of  these  coefficients  in  a  relatively  simple  form  involving  the  Bernoulli 
numbers. 


Special  Functions  35 

the  plane,  relation  (G)  continues  to  exist  provided  that  the  term  rj-iz  be  replaced 
by  —  7r/4z. 

Thus  in  summary  we  may  say  that  throughout  any  sector  (vertex  at  z  =  0) 
of  the  z  plane  tvhich  does  not  contain  portions  of  the  pure  imaginary  axis,  the  function 
F(z)  defined  by  (1)  may  be  developed  asymptotically  in  the  form 

wherein  the  upper  or  lower  sign  is  to  be  taken  according  as  we  are  dealing  with  a. 
sector  in  which  the  real  part  of  z  is  positive  or  negative. 

This  result,  which  is  at  once  seen  to  be  consistent  with  the  known  relation 
(2),  illustrates  in  simple  manner  the  way  in  which  asymptotic  developments  for 
a  given  function  may  be  ascertained,  at  least  in  some  cases,  by  means  of  the 
general  theorems  of  Chapter  I.  This  will  be  further  illustrated  in  what  follows> 
wherein  we  shall  eventually  consider  cases  of  much  greater  generality.^ 

20.  In  §  19  we  have  considered  asymptotic  developments  of  F{z)  (cf.  (1)) 
which  are  valid  in  sectors  situated  in  the  right  or  left  halves  of  the  z  complex  plane. 
We  proceed  to  show  how  the  same  method  may  yield  analogous  developments 
holding  for  the  upper  and  lower  halves  of  the  plane,  exception  being  made  natur- 
ally of  those  (pure  imaginary)  points  corresponding  to  the  values 

z  =  ±  (2?i  +  l)i;        n  =  0,  1,  2,  .  •  • 

at  which  F{z)  becomes  infinite. 

For  this  let  us  consider  the  function 

*(.)  =  F(i.)  =  t  (2„  +  \y.  _  ^.  . 

Regarding  z  at  first  as  real,  we  place 

1 


{2W  +   1)2  -  z2 

and  again  undertake  to  apply  Theorem  IV  with  m  =  0.     This  can  be  done  only 
in  case  ^z(w)  is  analytic  in  w  throughout  the  right  half  of  the  lo  plane.     How- 

8  In  the  special  instance  before  us  it  may  be  shown  that  ao  =  04  =  ae  =  •  •  •  =  0.     In  fact 
if  we  substitute  in  (7)  the  form  for  F{z)  given  by  (2)  we  obtain 


42Le'^^  +  l  J         2z    Ll  +  e^'^^J       z^  ^  z^  ^ 


where  the  upper  or  lower  sign  is  to  be  taken  according  as  the  real  part  of  z  is  positive  or  negative, 
and  this  relation  is  seen  to  be  true  when  02  =  04  =  oe  =  •  •  •  =  0.  It  is  to  be  noted,  however, 
that  in  general  if  a  function  is  defined  by  a  series  of  the  type  of  (1)  (cf.  (12))  no  formula  analogous 
to  (2)  is  at  hand.  The  indicated  method  for  determining  the  asymptotic  development  of  the 
function,  however,  remains  the  same,  thus  leading  to  cocfTicicnts  oo,  Oi,  02,  •••,  which  arc  in 
general  not  all  equal  to  zero. 


36  Determination  of  Asymptotic  Developments 

ever,  we  are  concerned  with  large  values  of  1 2 1 ,  and  whenever  1 2 1  >  1  it  is  evident 
that  ipz{w)  will  have  a  pole  of  the  first  order  within  the  indicated  region  at  the 
point  10  =  {z  —  l)/2  or  w  =  —  (z  +  l)/2  according  as  z  is  positive  or  negative. 

Let  us  first  consider  that  z  is  'positive.  We  proceed  to  apply  the  theorem, 
subject  to  the  remarks  of  §  11. 

Since  the  residues  r^,  r^'  of  the  functions 

at  w  =  /3  =  §(s  —  1)  are  respectively 

■K  TT 


2iz'        2i2(e""+  1) 
so  that 


'  P  2  *  ^ 


Uze""^'  +  1 
we  may  write  (at  least  when  x  >  \{z  —  1)) 

t^,  {2x  +1)2-22      ^''^  4:iz  e"''  +  1 

(9) 

/    ru^-l)-^         r--  \  dx  11 

+     e2   V  Jo  +  4._.He  )  (2."+D^^2  -  2  (2.+    1)2-.^  +  "^^•^> 

in  which  Cz  and  l^zCr)  are  obtained  by  changing  z^  to  —  s-  in  (4)  and  (5). 
But  from  elementary  considerations,  the  third  term  in  (9)  reduces  to 

i  1      (2+l)(2a:+l-z) 
4z  ^^  (z-  l)(2x+  1  +  2)' 

Whence,  upon  allowing  x  to  increase  indefinitely  we  obtain  (z  >  1) 
tbr  ^  _   ^  g""-  1  ,  1  I    1  ,     z+  1 

^V2j  4^.^  g-  r.  +   1  -^-  2(1   _   ^2)  -+-  42  ^Og  ^  _    ^ 

(10) 

_i  rr       1 1       1    ^y 

ijo    L (1  + 2^2/)2  -  2^      {l-2iyy-  z'^je^"^  -  1 
and  hence  (Theorem  V,  Chapter  I) 

^^^)  ^(^)^4r2.'^-+^l  +  2-^  +  2-+'--> 

where  62,  &4,  '  •  •  are  determinate  constants. 

This  result  may  now  be  generalized  to  all  values  of  z  belonging  to  a  sector  S 
(center  at  z  =  0)  lying  in  the  right  half  of  the  z  plane,  exception  being  made,  as 
already  indicated,  of  the  points  z  =  2n  +  1;  w  =  0,  1,  2,  •  •  •.  In  fact,  we  have 
but  to  suppose  \z\>  1  to  have  in  (10)  two  expressions  equal  for  positive  values 


Special  Functions  37 

of  z  and  each  analytic  throughout  S  and  hence  equal  for  all  values  of  z  in  the 
same  region.^  Moreover^  the  last  term  in  the  second  member  (like  the  two 
preceding)  is  readily  seen  to  be  developable  in  ascending  powers  of  I/2-,  thus 
leading  to  a  series  which,  in  the  sense  of  §  13,  represents  the  same  term  asymp- 
totically for  all  values  of  z  in  S. 

Likewise,  the  same  relation  (11)  is  found  to  hold  true  for  a  corresponding 
sector  in  the  left  half  of  the  plane,  exception  being  made  of  the  points 

z=  -  (2m  +  1);        n=  0,  1,  2,  ••• 

so  that,  having  replaced  z  by  —  iz,  we  may  say  in  summary  that  throughout 
any  sector  {vertex  at  z  =  ())  of  the  z  plane  ivhich  does  not  contain  portions  of  the  real 
axis,  the  function  F{z)  defined  by  (1)  may  be  developed  asymptotically  in  the  form 

This  result  is  again  seen  to  be  consistent  with  the  known  relation  (2).^° 
21.  Generalization  of  Example  1.     The  method  above  illustrated  for  deter- 
mining asymptotic  developments   is  in   general   applicable  to  functions  F{z) 
defined  by  series  of  the  form 

^  ^       ,froX(n)  +  piz) 

where  2^(2)  is  an  integral  function  of  z  and  where  \{n),  ij.{n)  are  functions  of  n 
such  that  Theorem  IV,  subject  to  the  remarks  of  §  11,  may  be  applied  to  the 
expression 

^'^''^      Mw)  +  p{z) 
in  order  to  find  for  a  given  value  of  z  the  sum 

Z/.(.T). 

We  observe  in  particular  that  by  taking  2^(2;)  =  z'^  (q  =  integer  ^  1)  the 
expression  F{z)  (or  the  sum  of  a  number  of  such  expressions)  comes  to  include  a 
wide  variety  of  functions  having  radial  clusters  of  polar  singularities  in  the 
neighborhood  of  the  point  z  =  co — a  characteristic  common  to  many  of  the 
more  important  functions  of  analysis. 

In  cases  where  fz{w)  cannot  be  considered  as  a  function  of  the  complex 

'  It  may  be  remarked  that  the  last  term  in  the  second  member  of  (10)  is  analytic  throughout 
S  {\z\  >  1)  because  the  improper  integral  involved  converges  uniformly  for  values  of  z  in  any 
sub-region  S'  of  S  whose  boundary  does  not  touch  the  boundary  of  .S'.  (Cf.  Oscood,  "Encyklo- 
padie  der  math.  Wiss.,"  II,  2,  §  6.) 

*"  In  view  of  the  same  relation  it  appears  from  (11)  that  in  the  present  simple  case  we  have 
bi  =  bi  =  b^  =  •  •  •  =0  and  that  the  symbol  ~  may  be  changed  to  =.     Cf.  note  8,  p.  35. 


38  Determination  of  Asymptotic  Developments 

variable  w  =  x  -\-  iy  but  is  continuous  in  the  real  variable  x  we  may  frequently 
determine  the  desired  developments  by  use  of  Theorems  I,  II  or  III  of  Chapter  I 
(subject  possibly  to  the  remarks  of  §  10).  The  manner  in  which  Theorem  I  may 
be  thus  used  will  be  shown  in  the  following  example  wherein  an  important 
type  of  function  F{z)  different  from  that  of  §  19  is  taken. 

22.  Example  2.     To  obtain  asymptotic  developments  for  the  function 

(12)  /■(.)  =  n[i+^:]. 

As  in  example  1,  this  function  may  be  evaluated  beforehand  and  takes  the 
form 


„ — T2 


(13)  ^(^)  =  S^" 

thus  furnishing  a  check  upon  our  subsequent  results. 
We  begin  by  writing 

(14)  log  F{z)  =  Elog  [l  +  -H  =  Hm  r  Elog  (x'  +  2^)  -  2  Elog  a:]  . 

n^l  L  ^*    J  x=ooL:>:=l  x=l  J 

From  §  12  we  have 

x-l 

-  2  Zlog  a:  -  -  2  log  r(a;)  =  -  log  2ir  -  2{x  -  ^)  log  x 

(15)  -1 

+  2x  +  coi(.r);  hm  a;i(a:)  =  0. 

2;=-)- 00 

We  proceed  to  apply  Theorem  I  (Chap.  I)  with  m  =  \  to  the  first  summation 
in  the  last  member  of  (14),  taking  for  this  purpose  f{x)  =  log  (x^  +  2-)  and 
supposing  for  the  present  that  z  is  real  but  different  from  zero.  The  theorem 
may  be  applied  since  the  series  (37)  (Chap.  I)  becomes 

fl.Or)    =    E     r     I  J72l0g  O^-'  +  -')   1  <P2(.t)dt, 

y=z  Jo       [  ^■^  J  x=y+t 

which,  as  in  (38),  may  be  written  in  the  form 

Z    j,=a:    I   ax  ]x=  +By 

and  is  therefore  convergent. 
Thus  we  have 

Elog  (a:2  +  2-)  =  i  log  (1  +  z')  +  fi.(l) 

(16)  '='  .. 

+  J      log  {X"  +  Z')dx  -  i  log  (.t2  +  z")  +  ^,{X). 

^*  See,  for  example,  Tannery,  I.  c,  §  121. 


Special  Functions  39 

Moreover, 


/ 


X 

log  (.1-2  +  z'^)dx  =  X  log  (a-2  +  s2)  -20^+22  arc  tan 


z 


But 


so  that  by  combining  relations  (14),  (15)  and  (16)  we  obtain 
log  F{z)  =  -  log  27r  -  ^  log  (1  +  z')  -  2z  arc  tan  ^+  2  -  12.(1) 

+  limrOi-  -  I)  logfl  +^j+  2zarctan^+  a;i(a-)  +  12.(.t)  J  . 

lim  (^  -  I)  log  (  1  +  ^  )  =  0;        lim  coi(.t)  =  0;        lim  Q,{x)  =  0, 

and,  supposing  at  first  that  z  is  positive,  we  shall  have  lim  2z  arc  tan  {x/z)  =  ttz. 
Therefore,  we  may  write  (z  real  >  0) 

(17)  log  F{z)  =  -  log  2x2+  7r2  -  §  log  (l  +  -2  j  +  2  (^1  -  z  arc  tan-  j  -  12.(1). 
On  the  other  hand,  if  z  is  negative  we  obtain 

log  F{z)  =  -  log  (-  2tz)  -  ttz  -  I  log  (^  1  +  ^  j 

(18)  .  1 X 

+  2  I  1  -  z  arc  tan-  1  -  12.(1). 

We  now  observe  that  the  expression  12.(1)  is  a  function  of  z  which  is  single 
valued  and  analytic  in  any  region  whose  boundary  does  not  cross  the  axis  of 
pure  imaginaries.  Whence,  within  any  region  Ai  situated  in  the  right  half  of 
the  z  plane,  equation  (17)  may  be  used,  while  similar  remarks  apply  to  equation 
(18)  for  values  of  z  pertaining  to  any  region  A2  in  the  left  half  of  the  plane.  More- 
over, if  the  boundaries  of  Ai  and  A2  are  not  tangent  to  the  pure  imaginary  axis 
at  00,  the  function  12.(1)  vanishes  Hke  l/z^  when  |z|=  ^  in  Ai  (or  Ao)  and  is 
developable  asymptotically  by  Theorem  V,  Chapter  I,  in  powers  of  l/z^  within 
this  region.  It  therefore  remains  but  to  apply  the  result  stated  in  §  15  in  order 
to  say  that  throughout  any  sector  {vertex  at  z  =  0)  of  the  z  plane  ichich  does  not 
contain  portions  of  the  pure  imaginary  axis,  the  function  F(z)  defined  by  (12)  may 
he  developed  asymptotically  in  the  form 

wherein  the  upper  or  lower  sign  is  to  he  taken  according  as  we  have  a  sector  in  which 
the  real  part  of  z  is  positive  or  negative. 


40  Determination  of  Asymptotic  Developments 

This  result  is  at  once  seen  to  be  consistent  wnth  the  known  relation  (13).^- 
23.  We  proceed  to  show  how  asymptotic  developments  for  the  F{z)  of  §  22 
may  be  obtained  which  will  be  valid  in  sectors  that  may  include  the  pure  imagi- 
nary axis.     For  convenience  we  shall  convert  this  problem  into  the  following: 
"  To  determine  asymptotic  developments  for  the  function 


(19)  F{iz)  =  $(2)  = 


n[^-:^]' 


which  shall  hold  good  throughout  certain  sectors  that  include  the  real  z  axis." 

Considering  at  first  that  z  has  a  fixed,  positive,  non-integral  value  >  1,  we 
proceed  (cf.  (14))  to  study  the  expression 


(20) 


H(z)  =  lim  r  Zlog  (x'-  -  2^)  -  2  Elog  a:]  , 

x=xi  L  ^=1  •'■=1  J 


in  which  we  agree  to  write  log  {x-  —  z-)  =  log  (s-  —  x-)  +  iri  whenever  x  <  z. 
Then  e'"^'^  =  ^{z). 

In  order  to  obtain  a  form  analogous  to  (16)  for  the  first  sum  here  appearing, 
let  us  place  <pziw)  =  log  (ic  +  z)  +  log  {iv  —  2),  in  which  it  is  understood  that 
the  function  log  {w  —  z),  considered  as  a  function  of  the  complex  variable  w, 
is  rendered  single  valued  throughout  the  right  half  of  the  z("-plane  by  means  of  a 
cut  extending  from  the  point  iv  =  z  vertically'  downward  to  the  point  iv  =  00 . 

We  shall  then  have 

a-— 1  x—1  x-l  x—1 

(21)  E  log  ix'  -z')  =  Y:  <Pz{x)  =  E  log  {x  +  2)  +  Z  log  ix  -  z) 
z=i  x=\  i=y  x=i 

and  we  may  at  once  apply  Theorem  IV  {m  =  0)  of  Chap.  I  to  the  first  sum  in 
the  last  member,  thus  writing 

Zlog(.r+2)  =  ^log(l  +  2)-fi.(l) 

(22)  """' 

+  J  log  {x  +  z)dx  -  i  log  (x  +  2)  +  n,{x), 

where 

(23)  Q.(x)  =  -z  J^    log  (^  ^^-:;r,  )  ^^^VZTJ  • 

The  second  sum,  however,  can  not  be  treated  in  the  same  manner  owing  to  the 
presence  of  an  essential  singularity  of  the  function  log  {iv  —  z)  at  the  point 
w  =  z.  In  this  case  a  related  method  yields  the  desired  result  as  we  shall  now 
show. 

1^  By  means  of  (13)  we  may  show  (cf.  note  8,  p.  35)  that  in  the  present  instance 

Oj  =  04  =  Oe  =  •  •  •  =0. 


Special  Functions 


41 


Let  us  take  for  this  purpose  the  integral  of  the  function 


(p{w)' 


f,(w)  =  log  (lo  -  z),         (p{w)  =  e"^"'"^  -  1 


with  respect  to  the  complex  variable  w  from  the  point  w  =  z  —  ij  (j  =  any 
real,  positive  value,  arbitrarily  large)  situated  on  the  right  side  of  the  above 
mentioned  cut,  around  the  open  contour  ABCDCEFGFHI  indicated  in  the 
following  figure,  the  integration  terminating  at  the  same  point  on  the  left  side  of 
the  cut,  and  it  being  understood  that  the  two  closed  loops  CD  and  FG  include 


Fig.  2 


respectively  the  points  w  =  2,  3,  4,  -  -  • ,  p  and  w  =  p  -j-  1,  p  -\-  2,  •  ■  -,  x  —  1, 
where  p  is  the  integer  for  which  2?  <  2  <  /^  +  1 ;  it  being  understood  also  that  the 
closed  curve  BCEFH  forms  a  circle  of  arbitrarily  small  radius  ^  with  center 
at  the  point  w  =  z. 

It  follows  from  (30)  of  Chapter  I  that  we  may  then  write 


x—l  /-» 

J2  log  {x  -  z)  =    j 

x=1  J  CD 


,  ^  dw  +    I 


dw, 


where  CD  and  GF  denote  respectively  that  the  indicated  integrations  take 
place  in  the  positive  sense  along  the  closed  contours  CD  and  GF. 
Whence,  we  have  also 

(24)  Z  log  (X  -  z)  =  log  (1  -  .)  +    f^j'']  dw  -    f^"^^  dw, 


in  which  C  indicates  an  integration  over  the  entire  contour  from  A  to  I,  while 
L  indicates  an  integration  over  the  open  loop  ABCEFIII. 


42  Determination  of  Asymptotic  Developments 

Let  us  now  replace  C,  as  may  evidently  be  done,  by  the  figure  in  part  rec- 
tilinear and  in  part  semicircular  Abcdefghijkl  whose  vertical  sides  (produced) 
pass  respectively  through  the  points  w  =  1,  w  =  x.  With  the  understanding 
that  the  radii  of  the  arcs  jih,  cde  are  each  equal  to  e,  we  have  now  but  to  refer  to 
the  processes  employed  in  §  8  to  see  that  by  taking  j  =  oo  we  may  write  (24) 
in  the  form 

Z  log  (.r  -  2)  =  i  log  (1  -  s)  -  fi_.(l)  +  J  log  (w  -  z)dw 

^^^^    '^'  ''  r   fziw) 

-  \  log  (.r  -z)-  fl_.(.r)  +  £(6)  -  y  '^^  dw, 

where  the  path  M  extends  from  w  =  \  to  to  =  x  over  the  curve  IFECx,  where 
Zoo  denotes  the  path  resulting  from  L  by  placing  j  =  co,  where  12_z(.'r)  is  the 
expression  obtained  from  (23)  by  replacing  s  by  —  2  and  where  E(e)  denotes 
an  expression  which  becomes  infinitesimal  with  e  and  may  therefore  be  at  once 
neglected. 
Now, 

(26)  J  ^^^  ^^  ~  ^^^^'^  ^  ^*^^^  ~  ^^  ^^^  {w  -  z)  -  w];:=i 

=  (z  —  1)  log  (1  —  2)  +  1  +  (a:  —  z)  log  (x  —  z)  —  x. 
Again,  we  may  write 

J   (p{ic)  2«     ^  ^  /      &  V  '       2-^1  J  10  —  z 

as  appears  by  an  integration  once  by  parts.     Whence, 


I 


<pi:w) 


die  =  log  [e-2-i(^-ij)  -  1]  -  log  [e-2"^'^  -  1]. 


Upon  placing  j  =  co  and  making  use  of  the  relation  g-^Tnz  _  ^  —  _  oie'"'  sin  ttz 
it  thus  appears  that  the  last  term  of  (25)  (the  coefficient  —  1  included)  is  equal  to 
—  log  (—  1)  +  log  (—  2*)  —  iriz  +  log  sin  ttz  =  log  2i  —  wiz  +  log  sin  ttz. 

Let  us  now  combine  relations  (20),  (21),  (22),  (25)  and  (26),  availing  our- 
selves also  of  (15)  and  of  the  facts  just  observed  concerning  the  last  term  of  (25). 
Noting  mutual  cancellation  of  terms  and  placing  a;  =  co  in  the  final  result,  we 
arrive  at  the  relation 

//(z)  =  log  sin  TTZ  —  log  27r  +  log  2^  —  iriz  —  \  log  (1  —  z^) 

+  z  log  J-^+ 2-12.(1)  -^^{l). 

If  we  now  write 

(1\  1  —  z  z—  1 

1-^j;  Z  log  j-q^  =  TTZZ  +  2  log  ^-J 


General  Integral  Function  43 

and  then  introduce  the  relation 

—  log  2t  —  log  iz  +  log  2i  =  —  log  irz 
we  may  therefore  write 

Hiz)  =  log  '^-h  log  (l  -  4)  +  (2  +  .  log^)  -  [12.(1)  +  fi_.(l)]. 

Similarly,  we  arrive  at  the  same  result  when  z  has  a  negative  non-integral 
value. 

Furthermore,  the  fun3tion 

0,(1)  +  1^,(1)  =  -^J  ^  log  [(i_ij,)3_,,J,-SV^ 

is  single  valued  and  analytic  throughout  the  portions  of  the  2;  plane  lying  to 
the  right  of  the  line  z  =  1  -{-  iy  and  to  the  left  of  the  line  z  =  —  1 -\- iy  while 
the  same  function  is  developable  asymptotically  by  Theorem  V,  Chapter  I, 
throughout  the  same  regions  in  the  form 

^2   -1-  ^4  -^  2«  -1  • 

Noting  that  for  the  function  $(2)  defined  in  (19)  we  have  $(—  iz)  =  F(z) 
where  ^(2)  is  the  original  function  (12),  and  recalling  also  that 

sin  7r(—  iz)       e"^  —  e""^ 


7r(—  iz)  2irz       ' 

we  may  say  in  conclusion  that  throughout  any  sector  {vertex  at  z  =  0)  of  the  z 
plane  which  does  not  contain  portions  of  the  real  axis,  the  function  F(z)  defined  by 
(12)  may  be  developed  asymptotically  in  the  form 


F{z) 


2wz 


-[^  +  f+> 


This  result  is  seen  to  be  consistent  with  (13). 

24.  We  proceed  to  the  following  more  general  problem: 

Example  3.     Given 


or 

p  =  integer  ^  1  ^^ 
p  ^  P  <  p+ 1 

according  as  0  <  p  <  I  or  p  ^  I.     To  determine  asymptotic  developments  for  F(z). 
"  We  adopt  the  familiar  notation  exp  x  for  e*. 


44  Determination  of  Asymptotic  Developments 

This  problem,  in  view  of  the  important  role  which  the  F(z)  thus  defined  plays 
in  the  modern  theory  of  integral  functions,  has  already  received  considerable 
attention.^^  Our  purpose  here  will  be  to  deduce  through  a  uniform  method  based 
on  the  fundamental  theorems  of  Chapter  I  the  known  results  together  w4th 
others  of  a  supplementary  character.^^ 

^Ye  shall  suppose  at  first  that  p  is  non-integral  and  >  1  (p  <  p  <  2'  +  1, 
p  =  integer  ^1).  Also,  for  the  present  z  is  to  be  regarded  as  having  any  fixed 
value  (real  or  complex)  except  one  of  the  following:  2^'^,  3^'",  4}'^,     •  -. 

The  method  then  requires  that  we  take  for  consideration  the  expression 
(cf.  (20)) 

Elog  (a:"- z)  -o-Zlog.T  +  EE-(  -^         I       (^  =  Ijp 

x=\  x=\  x=l  v=l  '^    X'*'     /     J 

in  which  the  value  to  be  assigned  to  log  {x"  —  z)  may  for  the  present  be  taken  in 
any  manner  consistent  with  the  equation  exp  log  (x'^  —  z)  =  x"  —  z. 

Then  exp  H{z)  =  F{z)  where  F{z)  is  defined  as  above. 

We  proceed  to  study  the  behavior  of  the  first  term  appearing  in  brackets 
in  (27)  when  x  is  large. 

Following  the  method  of  §  23,  let  us  place 

f,{w)   =   log  {W^  -  2), 

where  ic"  is  understood  to  be  so  defined  as  to  be  real  when  w  is  real  and  positive 
and  where  the  logarithmic  function  is  understood  to  be  rendered  single  valued 
in  w  throughout  the  right  half  of  the  u'-plane  by  means  of  a  rectilinear  cut  ex- 
tending from  the  point  w  =  2"  vertically  downward  to  the  point  w  =  co ,  the 
value  of  z<'  being  determined  in  accordance  with  the  following  conventions:  if 
2  =  r(cos  (p-\-  i  sin  (p)  then  z^  =  r''(cos  pep  +  i  sin  p(p)  subject  to  the  relation 
—  2ir  <  (p  ^  0.  The  function  fz{iv)  having  been  thus  defined  and  defined 
uniquely  for  every  value  of  w  whose  real  part  is  positive,  let  us  now  impose  for 
the  present  the  additional  condition  upon  z;  viz.,  real  part  of  z''  >  1;  i.  e., 
r"  cos  p(p  >  1.     Next,  let  us  consider  the  complex  integral 


X 


^'^''^dw;         <p{iv)  =  e^---  1 


■c<p{w) 

^*  See  Mellin,  Ada  Soc.  Sc.  Fennicae,  Vol.  29,  No.  4  (1900);  Barnes,  Philosophical  Trans., 
Vol.  199A  (1902);  Lindelof,  Acta  Soc.  Sc.  Fennicae,  Vol.  31  (1902),  p.  53;  Wiman,  Arkiv  for 
matem.,  Vol.  1  (1903),  p.  105;  Mattson,  "  Contributions  h  la  th^orie  des  fonctions  entieres  " 
(These,  Upsala,  1905);  Hardy,  Quarlerbj  Journ.  of  Math.,  Vol.  37  (1906),  pp.  146-172;  Ford, 
Annals  of  Math.,  Vol.  2  (2)  (1910),  pp.  11.5-127. 

"  It  may  be  observed  that  this  problem  differs  from  the  earher  more  special  ones  of  §§  19, 
20,  22  and  23  in  that  no  formulae  are  at  hand  analogous  to  (2)  and  (13)  by  which  we  can  predict 
beforehand  the  character  of  the  solution.  The  present  problem  therefore  illustrates  to  good 
advantage  the  value  of  the  methods  which  we  have  been  using. 


General  Integral  Function 


45 


taken  from  the  point  ic  =  z''  —  ij  (j  =  any  real,  positive  value,  arbitrarily  large) 
situated  on  the  right  side  of  the  above  mentioned  cut,  around  the  open  contour 
C  =  ABCBDEFGHGI  indicated  in  the  following  figure,  which  contour,  as  a 
result  of  the  above  condition  r''  cos  pip  >  1,  necessarily  includes  the  point  iv  =  z'' 
within  its  interior,  it  being  here  understood,  as  in  Fig.  2,  that  the  point  I  is  the 


Fig.  3 


one  on  the  left  side  of  the  cut  corresponding  to  A  and  that  the  closed  loops  BC 
and  GH  include  respectively  the  points  w  =  2,  3,  4:,  -  -  • ,  q  and  w  =  q-\-  \, 
g  +  2,  •  •  •,  {x  —  1)  where  q  is  the  integer  for  which  q  <  real  part  z''  <  q  -\-  V^; 
also  that  the  curve  DEF  forms  a  circle  of  arbitrarily  small  radius  ^  surrounding 
the  point  w  =  z''. 

Corresponding  to  relation  (25)  of  §  23  we  thus  obtain 

x—l  /• 

E  log  {X''  -  2)   =  I  log  (1  -  Z)   -   fi,(l)  +  log  (W'^  -  Z)dw 

-  \  log  (.1-  -  2)  +  12.(.t)  -    {^^^  dw 

where  M  indicates  an  integration  over  the  path  IGFEDBx,^'  where  L  indicates 
an  integration  over  the  path  ADEFI  in  which,  however,  the  points  A,  I  are 
now  supposed  to  be  taken  at  an  infinite  distance  along  the  cut,  and  where  ^zix) 
is  given  by  the  formula 

1^  In  case  real  pari  zp  =  q  =  an  integer,  the  indicated  loop  HG,  instead  of  containing  w  =  q 
in  its  interior,  will  have  this  point  upon  its  boundary.  To  obviate  the  difficulty  thus  arising, 
let  it  be  understood  in  this  case  that  the  cut  does  not  extend  vertically  do\\'nward  from  the  point 
w  =  ZP  but  first  extends  an  arbitrarily  small  distance  to  the  right  of  this  point  and  then  vertically 
downward  as  before.     The  reasoning  which  follows  will  then  apply. 

1'  In  case  zp  is  real  and  >  1  the  path  M  becomes  the  curve,  in  part  rectilinear  and  in  part 
semicircular,  \FECx  of  Fig.  2;  while  if  imag.  part  zp  <  0  the  path  M  may  be  taken  as  the  straight 
line  Ix  (Fig.  3). 


46  Determination  of  Asymptotic  Developments 

or 

(29)  ^^(-)=-^i    log[^^_.^^._J^,^, 

it  being  understood  that  the  integrand  of  (29)  is  so  defined  as  to  be  equal  for 
all  values  of  x  and  y  to  the  integrand  of  (28). 
Now,  an  integration  once  by  parts  shows  that 

/r   dw 
log  {w"  —  z)dw  =  IV  log  {w"  -  z)  -  aw  -  <^z  J  ^^  _  ^' 

Whence, 

r  C     dw         .      ^^        .    , 

log  (26"''  -  z)dw  =  X  log  (.1-'"  -  z)  -  ax  -  az  _     -  log  (1  -  2)  +  a. 


We  have  now  but  to  recall  the  formula  (15)  to  see  that  the  first  two  terms  in 
the  square  bracket  of  (27)  combine  into  the  following: 

Z  log  iw"  -  2)  -  0-  Z  log  .T  =  -7>  log  2tv  -\  log  (1  -  2) 

1=1  x  =  l  - 

(30)  -  fi^(l)  +  a;(.r)  +  a  +  (.r  -  §)  log  ( 1  -  ^;^) 


-x,^.+«.(^)-j; 


^(w) 


We  turn  next  to  consider  the  third  term  appearing  in  square  brackets  in  (27). 
By  use  of  the  well-known  relation^^ 

11  ,         1         ,    n^~'     ,    «  /  s  I  r^d^  V(irt  p>  0, 

!-»  =  1  +  21  +  3-,+  •■■  +  (;^3T)-.  +  (-^+''<W;       {h^  «,(„)  =  o 

n=oo 

wherein  f  represents  the  Riemann  f  function,  we  may  write 


(31) 


Um  ^^(a:)  =  0. 


Whence,  upon  observing  that  the  sixth  term  in  the  second  member  of  (30)  is 
of  the  form 

00  V 

-  Z);,.a.-i+  ^(-^O;        lim  77 (.r)  =  0 
i*See  Petersex,  "Vorlesimgen  iiber  Funktionstheorie"  (Copenhagen,  1898),  pp.  161-169. 


General  Integral  Function  47 

also  that  lim  Qz{x)  =  0,  we  arrive  at  the  following  relation  after  combining 
(27),  (30)  and  (31)  and  placing  x  =  co 

(32)    ^(2)=-^og27r-ilog(l-2)-fi.(l)+Er(^^)7  +  S(2)-  f^^.dw, 
where 

(33)  s(z)  =  1-m  4 1  +  E  (1  _  ,,),,.-.  -  ^  j,,,;;^^^  J  ■ 

The  properties  of  <S(2)  will  be  considered  in  further  detail  later.     For  the 
present  we  turn  to  the  last  term  of  (32). 
By  placing 

/.      /        X  1  /  N  7  ^^  d^ 

u  =  fz(io)  =  log  (iV'  -  z),         dv  =  ^^  =  ^2.i,^  _  1 
so  that 

<?w  =  -^ div,        V  =  :z~.  log  (e-^'^i^  -  1) 

it  appears  that 

f-^^  (iw  =  :^  log  (w"^  -  2)  log  {e-^"'^  -  1) 
J   ^(w)  27r^ 

27riJ 


(34)  ^     ^10"-^  log  (e--"^'^  -  1)  , 

aw. 


ifj" 


Now,  the  difference  between  the  value  of  the  first  term  here  appearing  on 
the  right  when  considered  at  the  point  lo  —  A  and  its  value  at  the  point  w  =  /  is 

(35)  log  [—  1  +  exp  —  2Tri  {real  'part  z''  —  ij)] 

as  appears  by  making  the  substitution  w"  =  s  oi  lo  =  s''  and  evaluating  the 
resulting  expression  between  corresponding  s  limits.  Moreover,  by  means 
of  the  same  substitution  the  second  term  in  the  right  member  of  (34)  becomes 


(36) 


]_   r  log  [-  1  +  exp  -  2Trisf] 

2ivi  J  s  —  z  ' 


where  the  integration  is  extended  over  a  contour  in  the  s  plane  which  includes 
no  singularities  of  the  integrand  except  the  simple  pole  aX  s  =  z.  But  the  value 
of  (36)  is  evidently  the  negative  of  the  residue  of  the  integrand  at  the  point 
s  =  z;  i.  e.,  —  log  [—  1  +  exp  —  2-Kiz'']. 

Since  the  expression  (35)  becomes  log  (—  1)  in  the  limit  as  j  =  oo,  it  thus 
appears  that 

(37)  f-^f '^  dw  =  log  (-  1)  -  log  [-  1  +  exp  -  2^2% 

If  in  (32)  we  place  log  (1  —  2)  =  log  (—  2)  +  log  (1  —  I/2;)  we  may  there- 


48  Determination  of  Asymptotic  Developments 

fore  write  the  original  function  F(z)  in  the  form 

(38)  F{z)  =  A{z)B{z), 

where 

■4(2)—  2p —  .2p —  2p, 

—  V—  z  V27r       —  I  V27rz''  V2xz'' 

and 


(39) 


B{z)  =  exp  [i:r(^.)^+  5(.)  -  12.(1)  -  i  log  (^1  -^)  J. 


Thus  far  we  have  supposed  z  to  have  any  value  (real  or  complex)  such  that 
2  ^  72,1 /p;  11=  i^  2,  3,  •  •  •,  and  such  that  having  placed  z  =  r(cos  (p -\-  i  sin  v?) 
and  agreed  to  write  z''  =  r''(cos  pep  +  i  sin  p^)  with  —  27r  <  ^  ^  0,  we  have 
r**  cos  p(p  >  1.  We  now  proceed  to  study  in  further  detail  the  expression  B{z) 
and  for  this  it  is  desirable  to  remove  for  the  present  all  restrictions  as  regards  2, 
thus  enabling  us  to  determine  certain  functional  properties  of  the  same  expression. 

We  turn  first  to  the  expression  ^.(l)  which  appears  in  B{z)  and  which  by 
reference  to  (29)  is  seen  to  be  defined  by  the  relation 


(40) 


^,,,  •  ^^      [i^  +  wr-zl      dy 


For  a  given  value,  real  or  complex,  of  z  this  12.(1)  evidently  has  a  meaning  unless  z 
be  such  that  the  equation  (1  ±  iyY  =  z  has  a  real  solution  in  y.  In  order  to 
determine  the  values  of  z  for  which  this  happens,  let  us  place 

z  =  r(cos  (p  -\-  i  sin  (p) 

and  make  the  conventions  already  indicated  as  to  the  meaning  of  z''.  For  the 
exceptional  values  in  question  we  must  then  have  1  ±  iy  =  r''(cos  pep  zt  i  sin  pep) 
so  that  the  same  values  are  those  lying  on  the  locus  of  the  equation  r*"  cos  pep  =  1; 
—  2t  <  ep  ^  0.  Whence,  if  r  be  large  the  same  values  will  tend  to  have  an 
argument  of  the  form  —  {2n  +  l)7r/2p  wherein  7i  is  a  positive  integer  for  which 
the  same  argument  lies  between  —  27r  and  0.  Again,  if  the  locus  just  mentioned 
be  drawn,  the  z  plane  is  thereby  divided  into  portions  in  each  of  which  12.(1) 
is  a  single  valued,  analytic  function  of  z,  since  within  any  sub-region  T'  lying 
wholly  within  such  portion  the  convergence  of  the  integral  in  (40)  is  readily  seen 
to  be  uniform.  Moreover,  if  arg  z  has  any  value  other  than  one  of  the  exceptional 
type  just  mentioned,  we  may  write    lim  12.(1)  =  0.     In  fact,  upon  reference  to 

Z  I  =00 

Theorem  V,  Chapter  I,  it  appears  that  under  such  hypotheses  we  shall  have 

where  the  coefficients  oi,  02,  •  •  •  may  be  obtained  by  expanding  the  integrand 
of  (40)  in  ascending  powers  of  1/z  and  integrating  term  by  term. 


General  Integral  Function  49 

Secondly,  we  turn  to  the  expression  S{z)  defined  by  (33).     Since 


w"  -z  ^  ^ivF'      w^^iw"  -  z) ' 
we  may  write 

r    dw       f.        z"  ^^  r       dw 


2)' 


Whence,  recalUng  that  M  extends  from  w  =  1  to  w  =  x,  we  obtain  the  following 
relation : 

where  N  represents  the  path  obtained  from  M  by  supposing  a;  =  +  oo . 

In  the  consideration  of  this  expression  we  have  thus  far  considered  that 
real  part  z''  >  1  and  from  what  has  already  been  noted  it  follows  that  if  we  have 
also  imag  part  s^  <  0  we  may  replace  (42)  by 


(43)  Siz) 


r^    z"        ^^^  r      dx      1 

''lhl-(Tv       ^'     X     x'-^ix'^-z)] 


The  form  of  (42)  may  also  be  simplified  when  i7nag  part  s*"  >  0  {real  part 
z*"  >  1).     In  fact,  we  may  then  write 

(44)      r ^j^ =  r ^ r ^^ 

where  the  last  symbol  represents  an  integration  in  the  positive  sense  about  the 
circle. 

Moreover,  the  last  term  of  (44)  is  readily  evaluated  and  found  to  be  equal 
to  2Tripz''-P-\ 

Thus,  when  imag  part  z"  >  0  (real  part  >  1)  we  may  write 


(45) 


sw-[|:j^„-.-f,-.,(^]+2.,v. 


These  facts  being  premised,  let  us  consider  the  properties  of  the  right  member 
of  (43),  all  assumptions  as  regards  2  being  laid  aside  for  the  moment.  Evidently, 
the  expression  in  question  represents  a  function  of  z  which  is  single  valued  and 
analytic  in  any  region  T  which  does  not  cut  the  portion  of  the  real  z  axis  extending 
from  z=lto2:=  +  oo.  Moreover,  when  1 2 1  <  1  we  find  upon  expanding 
in  ascending  powers  of  z  that  the  same  expression  is  developable  in  the  form 


(40)  <rE™=E  — 


Z_ 
V 


For  large  values  of   |z|  the  properties  of  the  right  member  of  (43)  are  now 
5 


50  Determination  of  Asymptotic  Developments 

derivable  by  means  of  the  following  Lemma  which  we  shall  state  and  prove  at 
this  point  and  to  which  we  shall  have  occasion  to  refer  frequently  throughout 
what  follows. 

Lemma}^     If  the  coeflBcient  g{n)  of  the  power  series 

,,„,  v^    /  V  a  =  integer,  positive,  neqative  or  zero 

(47)  Z  ^(n)z";  .        i.  ^  n 

„=a  r  =  radius  oj  convergence  >  0 

is  such  that  (o)  when  considered  as  a  function  g{ic)  of  the  complex  variable 
w  =  X  -{■  iy  it  is  single  valued  and  analytic  throughout  all  portions  of  the  w 
plane  lying  to  the  right  of  (or  upon)  the  vertical  line  lo  =  a  —  ^  -\-  iy  except  for 
a  finite  number  of  poles  situated  at  the  points  lo  =  Xi,  X2,  •  •  • ,  X*,  •  •  • ,  Xn ;  X(  =f  in- 
teger ^  a-°  and  (6)  is  such  that  to  an  arbitrarily  small  positive  quantity  e  there 
corresponds  a  positive  constant  K  (independent  of  x  and  y)  such  that 


g{x  ±  iy) 


9(.x) 


<  K  exp  ey 


for  all  values  of  .t  ^  a  —  ^  and  for  all  positive  values  of  y  sufficiently  large,  then 
the  function  f(z)  defined  by  (47)  when  1 2;  |  <  r  may  be  extended  analytically 
throughout  the  whole  z  plane  with  the  exception  of  the  positive  half  of  the 
real  axis,  and  throughout  this  region  will  be  defined  by  the  equation 

(48)  /(.)  =  -^J_^ ^;^- dy  -  Z  r, 

in  which,  if  we  place  z  =  r(cos  (p  -\-  i  sin  (p)  it  is  supposed  that  we  write 
(_  2)a-.i-f.  =  exp  [(a  -  i  +  iy)  log  (-  z)] 

=  exp  [(a  —  ^  +  iy)(\og  r -{-  i(p  -\-  irr)] 
and  take  —  27r  <  ^  <  0  and  in  which  r<  represents  the  residue  of  the  function 

(49)  ^g(^)(-  ^)'^ 

sin  TTW 
at  the  pole  w  =  X<." 

For  the  proof  of  this  lemma  let  us  at  first  suppose  for  simplicity  that  g{w) 
has  no  poles  at  the  right  of  (or  upon)  the  vertical  line  w  =  a  —  \  -\-  iy  and  let 
us  regard  z  for  the  present  as  having  any  fixed  value.  The  lemma  then  results 
from  a  consideration  of  the  result  obtained  by  integrating  the  function  (49) 

"  Cf .  LiNDELOF,  "Calcul  des  residus"  (Paris,  1905),  p.  109;  also,  Ford,  Journ.  de  Math., 
Vol.  9  (5)  (1903),  p.  223;  also.  Bulletin  of  Amer.  Math.  Soc,  Vol.  16  (2)  (1910),  p.  507. 

2"  This  condition  is  fulfilled  from  the  fact  that  g{n)  has  a  meaning  when  71  =  a,  a  +  1,  a  +  2, 
•  •  • .     Otherwise  the  given  series  (47)  would  lose  significance. 

2'  This  condition  is  satisfied  in  particular  if  constants  Ci  >  0  and  C2  ^  Ci  exist  such  that 
Ci  <  \g{u-)\  <CV 


General  Integral  Function  51 

about  the  rectangular  contour  C„  formed  in  the  iv  plane  by  the  lines 

w  =  a  —  ^  -\-  iy,        IV  =  ^  +  2n  -\-  iy,        w  =  a:  zb  ij 

where  n  is  any  integer  such  that  2n  >  a  and  where  j  is  any  positive  quantity, 
arbitrarily  large.  Upon  applying  (30)  of  Chapter  I  to  the  result  of  such  an 
integration,  we  arrive  in  the  first  place  at  the  relation 

f'a^                                   V    ^  ^  n        1    rg(^)(-z)"', 
(oO)  2^  5'(w)z"  =  W--  I    ■ dw. 

Supposing  at  first  that  z  is  real  and  negative,  we  proceed  to  study  the  integral 
here  appearing  in  further  detail. 

First,  along  the  side  of  Cn  upon  which  w  =  x  +  ij  we  have  dio  =  dx  and 
sin  TTiv  =  sin  7r(.T  +  ij)  =  sinh  7r;(sin  irx  coth  wj  +  i  cos  ttx)  so  that  if  we  call 
the  contribution  from  the  side  in  question  I,  we  may  write 

J  =    (-  ^y'    r~'-       g{x  +  ij){- zY       ^^ 


r 


2i  sinh  irj  .'tn+s,  sin  wx  coth  irj  +  i  cos  irx 
Whence  lim  7=0  provided  that 

j=ao 

(51)  lim  e-'^gix  +  ij)  =  0;        x  ^  a  -  ^. 

Similarly,  we  find  the  same  result  for  the  contribution  arising  from  the  side 
of  Cn  upon  which  w  =  x  —  ij  provided,  however,  that 

(52)  lim  e'^^gix  —  ij)  =  0;        x  ^  a  —  I. 

We  observe  that  both  conditions  (51)  and  (52)  are  satisfied  in  the  present  case 
as  a  result  of  (6)  of  our  hypotheses. 

Next,  let  us  consider  the  side  of  C„  upon  which  w  =  |  +  2/i  +  iy.  Here 
w^e  have  div  =  idy,  sin  iriv  =  cos  iiry  =  cosh  wy  so  that  having  taken  i  =  <» , 
the  contribution  in  question  becomes 


/=^- 


z)i+^  n  g(^  +  2n+iy){-z)'"_^ 

2  Xoo  COshTTW  ^' 


and  it  follows  from  (6)  of  our  hypotheses  that  the  improper  integral  here  appearing 
has  a  meaning  (z  real  and  negative).  Moreover,  it  follows  likewise  that  if 
l^l  <  r  we  shall  have  lim  J  =  0. 

n=oo 

Whence,  if  we  now  take  account  of  the  contribution  arising  from  the  remaining 
side  w  =  a  —  ^  -{-  iy  of  Cn,  noting  that  we  here  have  sin  ttw  =  (—  1)""^  cosh  iry 
while  the  integration  takes  place  from  ?/  =  +  °°  to  y  =  —  oo ,  we  may  write 

(53)  !:,(„),.  =  (=i)!  r.(a-»  +  «>)(-.r-'^-- 

^     ^  ^^^a"^  2      J_«,  coshx?/ 


52  Determination  of  Asymptotic  Developments 

This  relation  must  hold  good  as  we  have  indicated,  for  all  values  of  z  which 
are  real  and  negative  and  such  that  |z|  <  r.  But  the  first  member  represents  a 
function  of  the  complex  variable  z  which  is  single  valued  and  analytic  throughout 
the  circle  of  convergence  of  (47)  while  the  second  member,  with  the  conventions 
introduced  in  the  lemma  as  regards  the  meaning  of  (—  2;)°"^+'^  represents  a 
function  of  z  which  is  single  valued  and  analytic  throughout  the  whole  z  plane 
except  for  the  positive  half  of  the  real  axis.  In  fact,  for  all  values  of  z  in  a  region 
T  which  does  not  cut  or  touch  the  positive  half  of  the  real  axis  we  shall  have 
from  the  indicated  conventions  —  t  <  (p  <  w  so  that  upon  introducing  (6)  of 
the  hypotheses  it  appears  that  we  may  choose  e  so  small  that  the  improper 
integral  in  (53)  will  converge  uniformly  for  all  values  of  z  in  T.  Whence,  the 
same  integral  will  have  the  analytic  properties  just  indicated,  and  we  reach  in 
summary  the  lemma  for  the  case  in  which  g{w)  has  no  poles  to  the  right  of  the 
line  10  =  a  —  ^  +  iy. 

That  the  lemma  holds  true  in  the  more  general  case  follows  at  once  upon 
noting  that  relation  (50)  then  continues  {n  sufficiently  large)  provided  we  add 
to  its  first  member  the  expression 

TO 

Er,. 

Returning  to  the  second  member  of  (43)  which  is  defined  when  1 2 1  <  1  by  the 
series  (46),  let  us  now  apply  the  above  lemma  to  the  latter  series,  taking  for  this 
purpose  g(iv)  =  l/(p  —  iv).     Since  the  residue  of  the  function 

7r(-  zr 


(54) 


(p  —  w)  sin  irp 


at  the  pole  iv  =  p  is  —  7r(—  2) ''/sin  Trp,  it  thus  appears  that  for  all  values  of  z 
except  those  real  and  positive  we  may  write  the  expression  in  question  in  the 
form 

sm  Trp 
where 


«(p  +  ^-^y)cosh7r2/^^' 


it  being  here  understood  that  the  expressions  {—  zY  and  (—  2)""=+''^  are  to  be 
interpreted  in  accordance  with  the  conventions  stated  in  the  lemma,  i.  e.,  if 
2  =  r(cos  (p  -\-  i  sin  (p)  with  —  2ir  <  (p  '^  0,  then 

(—  2)"  =  exp  p(log  r  +  i(p  +  iir)  =  r''[cos  p(<p  +  ir)  +  i  sin  picp  +  tt)] 

and 

(—  2)-^+'"^'  =  exp  [(-  ^  +  iy){\og  r  +  i<p  +  iir)]. 

Upon  referring  to  (43)  and  (44)  it  follows  then,  as  regards  the  expression 


General  Integral  Function  53 

Siz)  originally  defined  by  (33),  that  throughout  any  region  Ti  of  the  z  plane  in 
which  real  part  z''  >  1,  imag  part  z''  <  0  we  shall  have 

(55)  S{z)  =  '^^^^+R{z), 
^     ^  sin  7rp 

while  throughout  any  similar  region  T2  in  which  real  part  z''  >  1,  imag  part  z"  >  0 
we  shall  have 

S(z)  =  ''':~^^'+  2Tipz'  +  R{z). 
sm  7rp 

Hence,  according  as  z  lies  in  Ti  or  T2  the  expression  B{z)  defined  by  (39)  takes 

the  form 

Biz)  =  exp  C(z)     or    B(z)  =  exp  [C(z)  +  27rip2;''], 
where 

c(z)  =  i;r(^)5+l^'-^iog(i-^)-o.(i)  +  ij(.). 

We  note  also  that  since  R{z)  is  equal  to  l/27ri  multiplied  by  the  result  of 
integrating  the  expression  (54)  from  ?/=  —  00  to  y  =  -\-  co  along  the  ^ine 
2V  =  —  ^  +  iy  it  follows  that  we  may  replace  R{z)  by  a  similar  expression  R{z) 
in  which  the  path  of  integration  is  w  =  —  k  —  ^  -\-  iy  (k  =  arbitrarily  large 
positive  integer)  provided  this  R{z)  be  increased  by  the  sum  of  the  residues  of 
the  function  (54)  at  the  poles  iv  =  —  1,  —  2,  -  ■  ■ ,  —  k.  Moreover,  since  these 
residues  form  the  first  k  terms  of  a  series  of  the  form 

(56)  ffo  +  —  +  -|  +  •  •  • ;        fli,  02,  •  •  •  constants  as  regards  z 

z        Z" 

while  lim  z'^'Riz)  =  0  it  follows  that  the  original  expression  Riz)  is  developable 

I  2  ,  =  <» 

asymptotically  in  the  form  (56)  (arg  z  4=  0). 

It  follows,  then,  upon  reference  to  (38)  and  to  the  properties  which  we  have 
now  estabhshed  for  12^(1)  and  R{z)  that  we  shall  have  the  following  relation  in 
which  the  upper  or  lower  of  the  double  sign  ±  is  to  be  taken  according  as  z  is 
confined  to  T2  or  Ti: 

Upon  observing  that  when  imag  part  2"  >  0  the  function  exp  iriz''  is  develop- 
able asymptotically  in  the  form  (56)  with  ctq  =  0,  oi  =  0,  •  •  •,  while  the  same 
is  true  of  the  function  exp  —  iriz"  when  imag  part  z''  <  0,  it  appears  that  the 
above  relation  may  be  simplified  into  the  following  holding  good  for  values  of  z 
in  regions  of  either  type  Ti  or  T2 : 

^,  ,       2sinxz''         [i-J^\-'',       i-zYl 


"4  Determination  of  Asymptotic  Developments 

This  relation,  as  we  have  noted,  holds  true  only  when  real  part  z''  >  I.  We 
now  proceed  to  determine  an  analogous  relation  for  any  region  Tz  in  which 
real  part  z''  <  1. 

If  this  assumption  be  made  at  the  beginning,  the  cut  in  the  iv  plane  falls 
entirely  outside  the  rectangle  bfgk  so  that  we  at  once  obtain  (32)  except  that 
the  last  term  of  the  second  member  is  lacking.  Moreover,  the  expression  S(z) 
takes  the  form  (55)  so  that,  upon  writing  log  (1  — 2)  =  log  (—  2)+log  (1— (l/z)), 
we  have 

H{z)  =  -  ^  log  27r  -  ^  log  (-  2)"  -  i  log  (l  -  i  )  -  fi.(l) 

(59) 

t^l       \pj   P  Sm  TTp 

and  hence 

1  rpfy\z'     7r(-  2)M 

(60)  F{z)  ~  ,^,  exp     E  r     -     -+     • • 

Before  summarizing  the  preceding  results  into  a  theorem  it  is  desirable  to 
note  certain  corresponding  results  which  may  be  obtained  when  z''  is  confined 
to  the  real  domain  (!,  +  «»)  — a  case  not  included  in  the  above  discussion. 

If  this  assumption  be  made  at  the  beginning  the  corresponding  Fig.  3  becomes 
that  represented  in  Fig.  2,  except  that  the  cut  extends  from  the  point  to  =  z'' 
instead  of  from  the  point  w  =  z.  Thus  we  obtain  equation  (32)  as  before  with 
S{z)  defined  by  (33)  in  which,  however,  the  path  M  is  now  understood  to  be 
IFECx  of  Fig.  2.  We  arrive,  therefore,  at  (38)  in  which  A{z)  is  defined  as 
before,  while  B{z)  is  defined  by  (39)  with  S{z)  given  by  (42)  wherein  N  repre- 
sents the  path  IFEC  +  <».  In  this  form  S{z)  is  now  developable  (as  was  (43)) 
when  1 2 1  <  1  into  the  form  (46)  from  which  we  find  as  before  that  unless 

(p  =  arg  z  =  0 

the  expression  S{z)  is  given  by  (55).  In  other  words,  S{z)  will  be  given  by 
(55)  when  <p  has  any  one  of  the  following  values  (for  which  z''  as  above  defined 
is  real  and  positive)  except  the  value  ^  =  0: 

27r         At         Qtt  2kir  2kT  ^ 

(61)  0, , , ''■■'-IT'         ~'T'>~2^- 

p  P  P  P  P 

Moreover,  when  cp  =  0,  S{z)  preserves  a  meaning  as  appears  from  (42),  provided 
that  z  =1=  1,  so  that  the  same  expression  may  be  obtained  from  (55)  when  z  ==  r 
is  large  by  placing  therein  z  =  r(cos  v?  +  i  sin  <p),  observing  the  indicated 
conventions  as  to  the  meaning  of  (—  zY,  (—  z)~'+'"  and  passing  to  the  limit  in 
the  resulting  expression  as  (p  approaches  the  value  zero  through  negative  values. 
Thus  it  appears  that  when  z"  is  real  and  positive — i.  e.,  when  ip  has  any 


er  ^  1 


General  Integral  Function  55 

one  of  the  values  (Gl)  —we  shall  have  relation  (57)  in  which  the  negative  sign  is 
to  be  taken  before  the  expression  iriz''  which  appears  in  the  square  bracket. 

Upon  noting  the  various  sections  of  the  z  plane  which  correspond  respectively 
to  regions  of  the  types  Ti,  Ti  and  T^,  we  thus  arrive  at  the  following 

Theorem  I.     Given  the  typical  integral  function  of  rank  p  {order  >  0): 

with  the  assumption  that  p  is  such  that  p  <  p  <  p  -\r  I. 

If,  having  placed  z  =  r(cos  <^  +  ^  sin  (p)  we  agree  that  z''  and  (—  zY  shall  be 
defined  respectively  by  the  equations 

gP  =  r''(cos  p<p  -\-  i  sin  p<p);         —  27r  <  (j?  ^  0 

(—  2)"  =  r''[cos  p{(p  +  tt)  +  2  sin  p{ip  +  tt)] 

then  for  values  of  z  of  large  modulus  and  lying  within  sectors  of  the  type 

-'^^Tr<<p<--^T;        A;  =  0,  1,2,3,  •••;        -  27r  <  <p  <  0 
2p  /p 

we  shall  have 

where  f  is  the  symbol  for  the  Riemann  ^function,  while  for  values  of  z  of  large  modulus 
and  lying  within  sectors  of  the  type 

-^^^<^<-^^t;        /.  =  0,  1,2,3,  ...;         -  2r  <  <p  ^  0 
2p  2p 


we  shall  have 


^l2^^z'' 
provided  (p  does  not  have  one  of  the  exceptional  values  0,  —  27r/p,  —  47r/p,  —  Gtt/p, 

Moreover,  for  the  exceptional  values  of  (p  just  mentioned  we  shall  have  when  |  z  \ 
is  large 

F(z)  ~    o„  exp     —  TTiz''  +  2^  i  I  -  I  — r  TT  —. . 

*'>/2^  L  .t^       \P/^  SlUTTpJ 

In  the  following  figure  the  sectorial  regions  indicated,  I  and  II,  represent 
those  in  which  for  large  values  of  |s|  the  first  or  second  of  the  above  forms 
holds  good  respectively,  while  the  dotted  lines  represent  the  special  directions 
along  which  the  third  form  ai)plies.     It  is  to  be  understood  that  the  last  radial 


56 


Determination  of  Asymptotic  Developments 


line  drawn  is  that  upon  which  (p  =  —  97r/2p,  but  that  a  complete  figure  would 
contain  all  similar  lines  upon  which 


tp  = 


2k -\-  1 
2p 


tt;        h  =  0,  1,  2, 


and     (p  ">  —  2t, 


the  scheme  of  alternate  division  of  the  plane  into  sectors  of  types  I  and  II  being 
carried  forward  up  to  and  including  the  last  sector  thus  obtained. 


Fig.  4 


Upon  noting  that  for  values  of  z  which  are  real  and  positive  {z  —  r)  we  have 

,   x(—  z)''               .      ,          cos  pir  -{-  i  sin  pir 
—  TTiz''  +  -:77 =  —  TTir''  +  irr'' z-zr~r =  Trr''  cot  px, 


sm  Trp 


sin  pT 


it  appears  that  the  above  theorem  is  consistent  with  certain  results  of  Hardy  to 
be  found  in  the  Quarterly  Journal  of  Mathematics,  Vol.  37  (1905),  page  158  (later 
corrected  on  page  373).  For  values  of  z  for  which  arg  z  =  (p  ^  0  the  theorem 
is  not  altogether  consistent  with  the  results  of  Barnes  in  the  Philosophical 
Transactions,  Vol.  199vl  (1902),  page  470,  since  an  equivalent  to  the  first  of  the 
forms  above  is  there  assigned  to  F{z)  for  all  values  of  z  such  that  ^  4=  0  ( |  s  1 
sufficiently  large).  It  is  to  be  observed  that  both  Barnes  and  Hardy  take  for 
discussion  the  function  F(—  z)  instead  of  the  F(z)  employed  above. 

25.  In  the  discussion  of  the  function  F(z)  of  §  24  we  have  thus  far  supposed 
p  <  p  <  p  +  1  where  p  is  any  integer  ^  1.  The  corresponding  results  for 
cases  in  which  0  <  p  <  1  may  now  be  readily  supplied,  it  being  understood 
that  F{z)  assumes  the  first  of  the  forms  given  at  the  beginning  of  §  24. 

Proceeding  as  in  §  24,  we  obtain  equation  (27)  as  before  except  that  the 
third  term  in  square  brackets  is  lacking.  Whence,  equation  (32)  continues 
except  that  the  terms  involving  the  function  f  are  absent,  while  instead  of  (33) 
we  have 


S(z) 


=  lim  0-     1  —  s      -^ 

^=00      L  JmIV     —  zJ 


Thus  it  appears  at  once  that  the  theorem  of  §  24  holds  true  lohen  0  <  p  <  1 


General  Integral  Function  57 

provided  that  the  term 

.7  .  7  ,  y=l         \PJV 

there  appearing  be  then  omitted. 

26.  We  proceed  to  consider  the  remaining  cases  —viz.,  those  in  which  p  =  p  = 
an  integer.  The  function  F(z)  is  then  defined  by  the  second  of  the  forms  appearing 
at  the  beginning  of  §  24. 

Equations  (27)  and  (30)  are  now  obtained  as  before,  but  instead  of  (31)  we 
write 

Moreover,  the  last  sum  here  appearing  is  evidently  of  the  form 
c  +  log  X  +  02  (.t);        lim  02(.t)  =  0 

2=00 

where  c  represents  Euler's  constant. 

Instead  of  (32)  we  thus  obtain  in  the  present  case 

H{z)  =  -  |log  27r  -  I  log  (1  -  2)  -  fi,(l)  +  Z  i{<Tv)  - 
(62)  ^  "='  " 


+  ccTZ^ -\- S{z)  -    i'^^dio, 
'^  ""  J™  L*^  "^  ^^  -  ^)  log  (  1  -  7)  +  ^-^  log  X 


where 

S{z) 
(63) 

The  last  term  of  (62)  may  now  be  evaluated  as  before,  leading  to  equations 
(37)  and  (38)  in  the  latter  of  which  A{z)  is  defined  as  before  while  B{z)  is  now 
defined  by  the  relation 

(64)     B{z)  =  exp  pi:  ^{<Tv)  ^'  +  c<jzP  +  S{z)  -  fi,(l)  -\\og(\-^\\ 

In  order  to  study  the  functional  properties  of  the  present  function  S{z)  we 
first  note  that 

(.1^  -  §)  log  (  1  -  -^  )  =  -  I:  -^-  +  e,{x) ;         lim  ^^{x)  =  0, 
'y  z^  _'-^     z"  ^^  2" 


58  Determination  of  Asymptotic  Developments 

also  that  instead  of  (41)  we  may  now  write 

^^  2"  .  .       ..   r       dw 


Whence, 
(65) 


r   dw      "^       2"        ,     -       ,   ^1  r     «^ 

J  ^^^^,  =  £ (1  _ ,,)^,^^-i + ^^ i«g ^^  +  ^^^ J  ,^(,^^-^^- 

ice, 

L  .=0 1  —  (TV  jN^yio"  —  2)  J 

Upon  expanding  {w{xo''  —  z)]~'^  in  ascending  powers  of  I/2,  supposing  for  the 
moment  that  I2I  <  1,  we  obtain 

-  z^'      -r^ — -^  =  -  psP  E  ^— [  =  P-''  log   1  -  2) 

=  pi7r2P  +  252^  log  (2  —  1). 
Whence,  under  the  present  hypotheses  relation  (55)  becomes  replaced  by 

S(Z)    =    Z  ~ <rzP  +  TTtzP  +  2^  log  (2  -   1) 

or,  since 

1  _  -  j  =  2P  log  2  -  Z  ^^  _^  l)^v-p+i 

=  2Mog2-|^^-f;^^^.;         |2|>1, 
we  may  write  when  1 2 1  >  1 
(66)  S{z)  = h  TrizP  +  2P  log  2  +  r(2), 

where  r(2)  is  an  expression  developable  asymptotically  in  the  form  (56). 

The  form  (66)  for  <S(2)  is  then  that  which  corresponds  in  the  present  case 
to  (55)  — i.  e.,  it  holds  for  values  of  2  confined  to  any  region  Ti  sufficiently  remote 
from  the  origin  throughout  which  real  part  2^  >  1,  imag  part  z^  <  0.  The 
corresponding  form  for  regions  T2  in  which  real  part  2^  >  1,  imag  part  2^  >  0  is 
obtained  (cf.  (44),  (45))  by  adding  2iriz'^  to  the  right  member  of  (66).  Thus, 
instead  of  (57)  we  reach  in  the  present  case 

(07)    F{.z)~^^^^\eriz''  +  'tAlY^+"^{c-l)  +  z'\oiz\ 
■\2irz^  L  v=\     \P  /  V        P  J 

where  0  =  0  or  0  =  1  according  as  z  is  confined  to  Ti  or  T^. 

Upon  observing  that  when  imag  part  2^  >  0  the  function  exp  iriz'^  is  develop- 
able asymptotically  in  the  form  (56)  with  a^  =  ai  =  a^  =  •  •  •  =  0,  it  appears 


General  Integral  Function  59 

that  (58)  becomes  replaced  in  the  present  case  by 

„ .  .       2sin7r2P        ["^  ^  f  v\z''  ,   z^  ^  ^  ,        1 

F{z)  -^    2  exp     Zn  -     -+-  c-  1    +2Plog2     . 

This  relation  holds  true,  then,  whenever  real  part  z^  >  1.  In  case  real  part 
2^  <  1  the  equations  corresponding  to  (59)  and  (60)  become  respectively 

H{z)  =  -  2^  log  27r  -  -  log  (-  z)^  -  i  log  (^1  -  ^^  -  12,(1)  +  mz^ 

+  |:r(^)^+^(c-l)  +  2Mog2+r(2), 

Finally,  in  case  z^  is  real  and  positive,  i.  e.,  in  case  (p  has  one  of  the  arguments 

27r  47r  Gtt 

~7'  ~7'  ~7'    '"' 

we  find  by  reasoning  analogous  to  that  at  the  close  of  §  24  that  S{z)  vAW  be  given 
by  (66)  and  hence  we  shall  have  (67)  in  which  ^  =  0. 

In  summar}^  we  arrive  then  at  the  following 

Theorem  II.-^  Given  the  typical  integral  function  F{z)  defined  in  Theorem  1 
together  with  the  assumption,  that  p  =  the  integer  p. 

For  values  of  z  of  large  modulus  lying  within  sectors  of  the  type 

4A;+3     ^      ^       4fe+l  \k  =  {),  1,  2,  3, 


2p  2p  [  -  27r  <  v?  <  0;         (p  =  arg  z 

we  shall  then  have 

F(z)  -^  , exp     Xn-)-+-{c-  l)  +  2Mog(-2) 

^2T{-zy         L-1     VP/^       P  feV        /J 

f  being  the  symbol  for  the  Riemann  ^function,  and  c  representing  Euler's  constant, 
while  for  values  of  z  of  large  modulus  lying  within  sectors  of  the  type 

_  4A;+1  4k-  1  U'  =  0,  1,  2,  3,  • . . 

2p      '^  <  "^  <  2p      ""  \-2w<<p^0;         <p=  argz 


we  shall  have 


„.  ,       2  sin  TTzP         f'v^'     / "  \  z"  .   2^ ,  1 

F(z)^    ,  exp     Er(-)-  +  -(c-l)  +  2Mog2    . 

V27r2P  L  i'=i     \  2^  /  »'       P  J 


Cf.  Mattson  (I.  c),  pp.  15-17. 


60  Determination  of  Asymptotic  Developments 

27.  It  will  be  observed  that  the  integral  function  F{z)  considered  in  §§  24-26 
is  of  order  >  0.  Barnes-^  has  also  considered  the  corresponding  problem  for 
certain  type  functions  whose  order  is  eqnal  to  zero,  but  we  shall  confine  ourselves 
to  the  case  treated  above. 

Asymptotic  Developments  of  Functions  Defined  by  Pow'er  Series 

28.  The  results  thus  far  indicated  in  the  present  chapter  are  but  indirectly 
applicable  to  the  determination  of  asymptotic  developments  for  functions 
defined  by  power  series.  This  subject,  however,  is  one  of  evident  importance. 
We  shall  now  point  out  a  general  theorem  in  this  field,  resulting  from  the  lemma 
of  §  24. 

Theorem  III.     If  the  coefficient  g{n)  of  the  poiver  series 

(68)  2Z^(w)2";         r  =  radius  of  convergence  >  0, 

may  he  considered  as  a  function  g{ic)  of  the  complex  variable  w  =  x  -\-  iy  and  as 
such  satisfies  the  following  conditions:  (a)  is  single  valued  and  analytic  throughout 
the  finite  w  plane  except  for  a  finite  nuviher  of  singularities  situated  at  the  points 
w  =  w\,  W2,  '  •  • ,  Wp,  none  of  which  coincide  with  one  of  the  points  w  =  0,  1,  2,  3,  •  •  • , 
and  (b)  is  such  that  to  an  arbitrarily  small  positive  constant  e  there  corresponds  a 
positive  constant  K  {independent  of  x  and  y)  such  that 


g(x  ±  iy) 


g(.x) 


<  K  exp  ey 


for  all  values  (real)  of  x  and  for  all  positive  values  of  y  sufficiently  large,  then  the 
function  f{z)  defined  by  (68)  (|sl  <  r)  will  he  such  that  for  all  values  of  z  lying  in 
any  sector  {center  at  z  =  0)  that  does  not  include  the  positive  real  axis  we  may  write 

,.Q,  -.,  ,  f^  g(-l)      g(-2)      g(-3) 

(69)  f{z)  ^  -l^rm ; -^ -^ •  •  •, 

OT=1  2  2  2 

where  rm  represents  the  residue  of  the  function 

wg{w){-  zr 


(70) 

Sm  TTW 

at  the  point  w  =  Wm- 

In  order  to  prove  this  theorem  we  observe  that  for  all  values  of  z  except  those 
real  and  positive  we  may  at  once  apply  the  lemma  of  §  24  with  a  taken  as  an 
arbitrarily  large  negative  integer:  a  =  —  /,  and  write 

(71)  Z  g{n)z-  =   i:  g{n)z-  +  f{z)  =  -  Z  r^  +  et{z), 

n=—l  n=—l  m  =  l 

23  See  Philosophical  Trans.,  Vol.  199A  (1902).  pp.  46G-468. 


Functions  Defined  by  Power  Series  61 

where  €i{z)  vanishes  to  as  high  an  order  as  the  {I  +  |)th  when  |  z  |  =  co .     Whence 
follows  the  indicated  result. 

For  example,  let  us  consider  the  function 

(72)  f{z)  =  E "  ;        P  4=  integer  <  0. 

Here  we  may  take 

q(w)  =  

and  the  residue  of  (70)  at  the  pole  iv  =  p  is  readily  found  to  be 

7r(-  zY 


n 


sin  Tp 


Whence,  throughout  any  sector  such  as  indicated  in  the  above  theorem  we 
shall  have 

(73)  f{z)  ~  -  -^^y  -  l^^i^,  -  (^  +  2).^       •  •  ■  • 

This  result  ceases  to  hold  when  p  =  a  negative  integer  since  the  expression 
then  has  a  pole  of  the  second  order  at  w  =  p.  Such  cases  may,  however,  be 
treated  by  the  same  theorem.  Thus,  in  particular,  when  p  =  —  1  we  obtain 
directly 

log  (-  2) 
ri  = 

—  z 

and  hence,  instead  of  (73) 

,,  ,       log  (-   2)       1        1         1 

(74)  /(^)-^V--^^-2?-3^---- 

This  result  may  be  verified  by  noting  that  when  p  =  —  1  the  equation  (71) 
gives 

j^r  \       log  (1-  2) 

m  =  — ^ — 

while  the  power  series  appearing  in  (74)  converges  when  |  s  |  >  1  to  the  value 
I  log  (1  —  1/z)  so  that  (74)  gives  the  same  form  for/(z). 

29.  Generalizations  of  Theorem  Ill.—li  for  a  given  series  (68)  the  function 
giw)  is  not  single  valued  throughout  the  w  plane,  but  contains  q  branch  points 
w  =  'Wi,  ibi,  •  •  • ,  Wq,  conditions  (a)  and  (b)  remaining  otherwise  the  same,  the 
theorem  continues  true  provided  that,  after  rendering  giiv)  single  valued  by 
means  of  q  cuts  extending  vertically  downwards^'^  to   infinity  from  the  points 

2^  Since  the  series  here  appearing  is  convergent  for  \z\  >  I  tlic  symbol  ~  may  be  changed 
to  =. 

25  More  generally,  in  any  direction  tending  to  infinity  in  the  right  half  of  the  plane  or  vertically 
upwards  or  downwards. 


62  Determination  of  Asymptotic  Developments 

w  =  Wm',  (w  =  1,  2,  •  •  ■ ,  q)  respectively,  we  subtract  from  the  second  member  of 
(69)  the  expression 

9 

m=l 

where  am  represents  the  loop  integral  (assumed  to  exist)  of 

1  g{w){-  zr 
2i      sin  irw 

taken  in  the  positive  sense  from  the  point  w  =  Um  —  ioo  to  the  same  point  after 
surrounding  the  (one)  branch  point  w  =  fCrn-  This  result,  in  fact,  appears 
directly  upon  reference  to  the  demonstration  of  the  theorem. 

We  note  that  in  case  the  point  w  =  I'Cm  coincides  with  a  point  of  the  type 
IV  =  u'm  mentioned  in  the  theorem,  the  corresponding  value  of  rm  is  to  be  neglected, 
the  term  am  then  being  evidently  the  only  one  of  the  two  to  be  retained. 

A  particular  type  of  function  f{z)  to  which  Theorem  III  and  these  supple- 
mentary remarks  apply  is  the  following,  discussed  by  Barnes  :^^ 

f  r     fl\  _  V  ^''x(^^  +  d)  0  =  constant  #=  0  or  neg.  integer, 

h  (2;  ^J  -  £.    (n  +  ey    '         ^  =  constant, 

where  x(l/2)  is  regular  at  the  origin.  Besides  this,  Barnes  considers  the  corre- 
sponding problem  for  certain  special  types  of  functions  for  which  condition  (b) 
of  Theorem  III  is  not  fulfilled.  Of  these  latter  may  be  especially  mentioned 
the  function 

„  y^  2"  6  =  constant  4=  0  or  neg.  integer, 

J^^{z;d)  -  Z,  (^4_0)3r(,,_^  1) ;        ^  =  constaiit, 

for  which  it  is  stated-"  that  for  all  values  of  z  of  large  modulus  we  may  write 

F,(z;  d)  ^  <p,{z,  6)  +  e^z-'^ao+^  +  ^+  •••], 

where  ^3(2;  6)  represents  the  loop  integral  of  the  function 

_  J^  2T(-  s) 

2«  {s  +  ey 

taken  over  the  path  in  the  s  plane  extending  from  the  point  5  =  —  00  -f-  imag 
part  {—  6)  to  the  point  s  =  —  6  and  return.-^  The  values  of  ao,  Ui,  az,  •  •  •,  are 
also  given. 

This  function  F^{z',  6)  typifies  an  important  class  of  functions,  viz.,  those 
which  for  an  appropriate  value  of  arg  z  become  infinite  like  e'z''  {k  =  const.) 

"  Philosophical  Transactions,  Vol.  206A  (1906),  pp.  257,  272,  282. 
"  L.  c,  p.  265. 

^*  Barnes  examines  in  further  detail  the  properties  of  this  loop  integral,  expressing  it  in  the 
form  of  a  series  in  his  final  result  (p.  265).     Cf.  also  Quarterly  Journ.  of  Math.,  Vol.  37  (1906),  p. 
89  et  seq;  also  ibid..  Vol.  38,  p.  116  et  seq. 


Functions  Defined  by  Power  Series 


63 


when  \z\  is  large.  In  this  connection  the  following  more  general  statement 
seems  probable,^'^  though  a  rigorous  proof  of  it  cannot  be  supplied  by  the  author 
at  present. 

"  If  the  function  g(n)  appearing  in  the  coeflBcients  of  the  power  series 

may  be  considered  as  a  function  g(iv)  of  the  complex  variable  w  =  a:  +  iy  and 
as  such  satisfies  the  following  two  conditions  (o)  is  single  valued  and  analytic 
throughout  the  finite  w  plane  except  for  a  finite  number  of  singularities  situated 
at  the  points  w  =  Wi,  iV2,  •  •  • ,  Wp  (none  of  which,  however,  coincide  with  one 
of  the  points  w  =  0,  1,  2,  3,  •  •  •)  and  (6)  is  such  that  there  exists  a  constant 
(real  or  complex)  jS  for  which  the  function 

-  r(-  w) 


Giw)  = 


is  developable  in  the  form 


T{\-w-  /3) 


g{y^) 


(75) 
where 


^^''^  ~  2(;  +  /3"^  (w  +  ^)(w  +  /3  +  1)  "*"  {10  +  /3)(«^  +  /3  +  \){w  +  i3  +  2) 


+  •••  + 


hn  +  (^n{w) 


(w  +  mw  +  ^  +  1)  +  •  •  •  (w  +  /3  +  n) ' 


hm  en{io)  =  0;         —  9  =  (^W  lo  -^^ 

111-1=00  ■^  -^ 

then,  for  all  values  of  z  of  large  modulus  we  may  write 


-E^n.+    (-2)V|    60  +  7+1  + 

TO=1 


in  which  Vm  represents  the  residue  of  the  function 

=  -  r(-  io)g{w)i-  z) 


\ 


Tg{w){-  zY 


T(iu  +  1)  sm  TTW 

at  the  point  w  =  Wm  and  in  which  the  coefficients  60,  61,  62, 
from  (75). "30 

29  From  considerations  based  upon  the  relation 

2i  J    T(w  +  k)  sin  tw       2*-^  '^^\z/' 


are  determined 


k  =  constant  ^  1, 
P  {  -  )  =  polynomial  in  -, 

where  the  indicated  integration  takes  place  in  any  closed  (infinite)  contour  embracing  the  points 
w  =  0,  1,  2,  3,  •••. 

'0  No  mention  has  been  made  in  the  present  Chapter  of  a  class  of  power  series  whose  asymp- 
totic forms  have  been  studied  by  Dienes  and  Valikon.  For  a  concise  statement  of  their  results 
see  Theorems  I  and  II  in  Valiiion's  paper,  "Sur  le  calcul  approch6  de  certaines  fonctions  enticres, " 
Bull,  de  la  Sac.  Math,  de  France,  Vol.  42  (1914),  pp.  252-264. 


CHAPTER  III 

THE  ASYMPTOTIC  SOLUTIONS   OF  LINEAR   DIFFERENTLAL  EQUATIONS 

30.  The  oldest  and  most  fully  developed  aspect  of  the  theory  of  asymptotic 
series  concerns  the  so-called  "  asymptotic  solutions  "  of  linear  differential  equa- 
tions. In  the  present  chapter  we  shall  undertake  to  give  a  summary  of  the 
principal  results  (without  proofs)  that  have  been  obtained  in  this  field,  with 
indications  as  to  certain  noteworthy  questions  still  remaining  unanswered. 
Corresponding  results  and  questions  for  linear  difference  equations  will  also  be 
briefly  considered. 

Real  Variable 

31.  Confining  the  attention  at  first  to  the  case  in  which  the  independent 
variable  x  is  real  and  positive,  the  investigations  referred  to  may  be  said  to 
cluster  about  the  homogeneous  linear  differential  equation 

(1)  2/(">  +  ai(.r)y("-"  +  a2(.T)2/^"-'^  +  •  •  •  +  an{x)y  =  0, 

wherein  the  coefficients  ai,  a^,  •  •  •,  ««  are  assumed  to  be  developable  for  large 
positive  values  of  x  either  in  convergent  or  asymptotic  series  of  the  form 


arix) 


r  ,     flr.  1    I     ar.2     .  "I  ^      ^ 

ttr,  o+~+"^+---h        r=l,  2,  •••,n. 


h  being  zero  or  some  positive  integer.^  In  this  equation  the  point  .r  =  qo  is 
in  general  a  so-called  "  irregular  point  "^  so  that  the  usual  "  normal  solutions  " 
about  the  point  x  =  co ,  as  provided  by  the  well-known  theories  of  Fuchs, 
come  to  involve  power  series  in  \jx  that  are  divergent  for  all  values  of  x.^  Never- 
theless, the  same  solutions  continue  to  satisfy  the  equation  formally^  and  it 
can  be  shown  that  they  represent  asymptotically,  in  the  precise  sense  of  §  13, 
certain  actual  solutions.  In  fact,  we  may  begin  by  citing  the  following  note- 
worthy theorem  first  established  rigorously  by  Horn:^ 

"  If  for  the  equation  (1)  the  roots  ??u,  W2,  •  •  •,  7n„  of  the  characteristic  equa- 
tion—  i.  e.,  the  algebraic  equation 

'  The  integer  fc  +  1  is  termed  the  rank  of  (1)  at  a;  =  <x> .     See  for  example  Horn,  "Gewohn- 
liche  Differentialgleichungen  bcheber  Ordnung"  (Leipzig,  Goschen,  1905),  p.  187. 

2  For  an  exposition  of  the  definitions  and  basal  theorems  in  the  theorj^  of  linear  differential 
equations,  one  may  consult  Picard's  "Trait6  d'Analyse"  (1896),  Vol.  3,  Chap.  11. 

3  Cf.  PiCARD,  I.  c,  §  22. 
*  Cf.  PiCARD,  I.  c,  §  23. 

«  Cf.  Acta  Math.,  Vol.  24  (1901),  p.  289. 

64 


Real  Variable  65 

(2)  m"  +  ai,  om"-i  +  •  •  •  +  On.  o  =  0, 

are  distinct  from  one  another,  equation  (1)  possesses  n  linearly  independent 
solutions  yi,  y^,  •  •  • ,  yn  valid  for  large  positive  values  of  x  which  are  developable 
asymptotically  in  the  forms 

(2')  yr  ~  e'^'^h^^  E  ^" ;        r  =  1,  2,  •  •  • ,  n, 

where /r(a:)  is  a  polynomial  of  degree  ^*  +  1  in  x,  the  coefficient  of  whose  highest 
power  in  x  is  mrjik  +  1),  while  pr  and  Ar,  j  are  constants^  with  Ar,  o  =  1."^ 

If  in  this  theorem  the  restriction  be  removed  that  the  roots  of  the  charac- 
teristic equation  be  distinct — i.  e.,  if  multiple  roots  be  present — the  theorem 
fails  and  we  at  once  encounter  a  problem  for  which  no  general  solution  has  as 
yet  been  obtained.  However,  Love^  has  recently  made  a  noteworthy  advance 
in  this  direction,  his  theorem  (which  manifestly  contains  the  above  as  a  special 
case)  being  as  follows: 

If,  other  conditions  remaining  as  stated  above,  "  the  characteristic  equation 
has  I  roots  mi,  TO2,  •  •  • ,  mi,  occurring  respectively  rii,  ri2,  •  •  • ,  ni  times  (rii  +  na 
+  '•'-}-  ni  =  n)  and  such  that  no  multiple  root  of  the  characteristic  equation 
is  also  a  root  of  the  equation 

(3)  ai,  im"-i  +  a2.  im"-^  + h  fln.  i  =  0, 

then  the  equation  (1)  possesses  a  fundamental  system  of  solutions  yr,q  (r  =  1,  2, 
' '  ■ ,  I;  q  =  1,  2,  ' ' ' ,  Ur)  developable  asymptotically  in  the  form 


rir—l  00  J 


where  fr.qix)  is  a  certain  polynomial  of  degree  nr(k  +  1)  in  .r'"'',  the  quantities 
Pr,q  and  Ar,  q,  i,  j  arc  determinate  constants,  and  Ar,  q.o,o  —  !•" 

Love  has  furthermore  considered  in  detaiP  the  equations  (1)  of  the  second 
and  third  orders,  including  the  cases  in  which  (3)  is  satisfied  by  a  multiple  root, 

^  The  precise  values  of  the  coefficients  of  frix)  and  of  the  constants  pr,  Ar.j  may  be  deter- 
mined by  the  method  of  undetermined  coefficients  after  substituting  y^  in  (1).  A  similar  remark 
should  be  understood  with  reference  to  the/r,<,(a;),  pr.q,  etc.,  that  follow. 

^  Historically,  the  first  form  of  equation  (1)  to  be  studied  in  this  connection  was  that  taken 
by  Poincar6  in  which  Ci,  oj,  •  •  •,  a„  are  rational  fractions,  thus  possessing  no  other  singularities 
than  poles  at  x  =  «=.     See  Ada  Math.,  Vol.  8  (1886),  pp.  295-344. 

8  Cf.  Annals  of  Math.,  Vol.  15  (1914),  p.  155. 

"  Love  does  not  use,  at  least  directly,  the  method  common  to  the  greater  part  of  Horn's 
work,  viz.,  that  of  successive  approximations,  though  the  latter  could  doubtless  be  employed  to 
the  same  ends.  His  method  rests  rather  upon  certain  general  studies  of  Dini  to  be  found  in  Vol.  2 
(1898)  of  the  Annali  di  Mat.,  pp.  297-324,  wherein  the  equation  (1)  of  the  nth  order  is  first  con- 
verted into  a  VoLTERRA  integral  equation  of  the  second  kind  containing  n  arbitrary  functions, 
termed  "auxiliary  functions,"  and  the  latter  (equation)  solved  by  the  usual  process  of  iteration, 
thus  yielding  forms  of  solution  for  the  original  equation  (1).     Tlirough  the  arbitrariness  existing 


66  Asymptotic  Solutions  of  Differential  Equations 

and  has  arrived  at  complete  results  for  these  orders.^"    Thus,  for  n  =  2  we 
have  the  following  :^^ 

"  In  the  differential  equation 

2/"  +  h{x)y  =  0 

suppose  that  b{x)  is  a  real  or  complex  function  developable  asymptotically  for 
large  real  positive  values  of  x  in  the  form 


Hx) 


..«[^.  +  ^+...], 


where  )!:  is  0  or  a  positive  integer.  Then,  for  the  same  values  of  x  equation 
possesses  two  hnearly  independent  solutions  yi,  y^  such  that  (a)  if  ho  =t=  0,  i.  e., 
if  the  roots  mi,  m%  of  the  characteristic  equation  m^  +  &o  =  0  are  distinct,  we 
may  write 

^,  ~/'^'>a:''^[l+^'+  •••],         r=  1,2, 


where 


TiirX^^^  ,   ar.-fca:^ 


/'■(^)  =  ^ipT  "^  ^ !-•••+  «r.-l.T; 


1,2, 


(Jb)  if  6o  =  0,  6i  4=  0  we  may  write 

2/.^^'-^vTi+^^+--- +4(^^.0+%-+ • 

where 

(c)  if  ^-  =  6o  =  ^1  =  0  we  may  write  in  general 

yr'^x''^^l+^+  •••],         r=  1,2; 

(d)  but  if  p2  =  pi  or,  in  general,  if  p2  —  pi  is  a  positive  integer  we  have 

in  the  choice  of  these  auxiUary  functions,  the  resulting  solutions,  though  frequently  complicated, 
are  of  great  flexibility  and  it  thus  becomes  possible  to  adapt  them  to  a  wide  variety  of  investi- 
gations, as  DiNi  himself  has  abundantly  shown  in  a  series  of  papers  in  the  Annali  di  Mat.  extending 
over  the  years  1898-1910.  In  the  case  of  studies  such  as  are  being  considered  in  the  present  chap- 
ter, the  method  readily  provides  actual  solutions  that  are  valid  for  large  (positive)  values  of  x 
and  thus  the  problem  becomes  merely  that  of  showing  that  the  auxiliary  functions  may  be  chosen 
in  particular  in  such  a  way  that  these  solutions  are  developable  asymptotically  in  the  sense  of  §  13  . 

10  Cf.  Am.  Journ.  of  Math.,  Vol.  34  (1914),  pp.  165-166. 

"  For  the  sake  of  completeness  the  case  of  unequal  roots,  though  covered  by  the  above 
mentioned  theorems,  is  included  in  the  statement. 

12  It  will  be  obscr\'ed  that  (6),  (c)  and  (d)  relate  to  the  cases  in  which  ?«i  =  W2.  If  in  (c) 
or  (d)  the  series  for  bix)  converges  for  all  \x\  >  R  then  x  =  <»  is  a  "  regular  point  "  of  the  differ- 
ential equation  and  hence  in  the  results  for  yi  and  2/2  the  sign  fo  may  be  changed  to  = ;  lx|  >  R. 


12 


Real  Variable  67 

2/2  ~  2/1  log  X  +  x""-    yl2, 0  +  -^  +  •  •  •     •" 

The  complete  result  for  the  equation  of  the  third  order  is  as  follows : 
"  In  the  differential  equation 

y"'  +  h{x)y'  +  c{x)y  =  0 

suppose  that  h{x)  and  c{x)  are  real  or  complex  functions  developable  asymp- 
totically when  X  is  large  and  positive  in  the  forms 


where  A;  is  0  or  a  positive  integer,  and  suppose  that  h'{x)  also  has  an  asymptotic 
development.     Then  for  the  same  values  of  x  the  given  equation  has  three  linearly 
independent  solutions  yi,  yi,  yz  possessing  asymptotic  developments  as  follows: 
(a)  If  the  roots  mi,  mi,  mz  of  the  characteristic  equation 

m^  +  hum  +  Co  =  0 
are  distinct,  we  may  write 


where 


,/A)^P.[^14.^i_|_  ...J.         r=  1,2,3, 


irirX^'^^         ar,-kX^ 


(6)  If  Ml  4=  m2  =  Ws  we  may  write  in  general 

where /i(.r)  has  the  same  form  as  in  (a)  and 

(c)  But  if  in  (6)  ps  =  po,  or  in  general  if  pa  —  P2  is  a  positive  integer,  we  have 


68  Asymptotic  Solutions  of  Differential  Equations 

2/3  ~  2/2  log  X  +  e^'^'h''  \Az,  o  +  -^'  +  ' ' '  J  ' 

where /i (a:)  aiid/2(.r)  have  the  same  form  as  in  (a). 

(d)  If  7?zi  =  7712  =  W3   and   either  Ci  ^  0  or  6i  =  Ci  =  0,   C2  =[=  0  we   may- 
write 

where 

(e)  If  c\  =  0,  6i  =}=  0,  2/1,  2/2,  2/3  h^ve  expansions  of  the  same  form  as  in  (6). 
(/)  If  I'  =  6i  =  Ci  =  C2  =  0  we  may  write 


^2/1  log  a:  +  x^'     ^2.  o  +  -|^  +  •  •  •     , 

£2/1  log2  .T  +  .T^Uog  a:  [ 53. 0  +  ^^  +  •  •  •  ]  +  .T"'  [ ^3,  0  +  ^  +  •  •  •  ]  . 


13 


While  the  complete  results  for  the  equation  (1)  of  order  n  ^  4  have  not  as 
yet  been  obtained,  a  careful  examination  of  those  just  given  for  7i  =  2,  3  throws 
light  upon  what  the  corresponding  forms  may  be  expected  to  be.  Moreover, 
in  connection  with  this  question  the  following  result  should  be  noted:" 

"  Let  T(.r)  be  one  of  the  system  of  functions 

^^^'  .^'"     .T(logx)i+'"     .T  log  a:(log  log  .r)'+'"     *"'     (''  >  ^^ 

and  put 

Xoo 
T{x)dx. 

1'  It  will  be  observed  that  (6)  and  (c)  refer  to  the  case  ?«i  4=  w?2  =  m%  while  (d),  (e)  and  (/) 
refer  to  the  case  m\  =  m-i  =  mz.  If  in  (/)  the  series  for  h{x)  and  c(x)  converge  for  all  lx|  >  i? 
then  X  =  00  is  a  regular  point  of  the  differential  equation  and  hence  in  the  results  for  yi,  7/2  and  yz 
the  M  may  be  changed  to  =;  |x|  >  R. 

"  Obtained  by  Dini  for  the  case  in  which  the  roots  of  the  characteristic  equation  all  have  the 
same  real  part,  and  partially  obtained  by  him  when  this  restriction  is  removed  {Annali  di  Mat., 
Vol.  3  (1899),  p.  136.  The  result  has  recently  been  established  in  its  entirety  by  Love  in  the 
American  Journal  of  Mathematics. 


Complex  Variable  69 

Suppose  now  that  in  the  differential  equation 

(4)  2/(")  +  [ai  +  ai(.T)]2/(«-i)  +  [a^  +  «2(a:)]2/("-2)  +...  +  [«„  +  a„(y)]y  =  q 

the  functions  ai{x),  a2(x),  --  -,  an(x)  together  with  their  2n  —  1  derivatives  are 
continuous  when  x  is  sufficiently  large,  and  suppose  that  the  characteristic 
equation 

(5)  M"  +  aiM""'  + h  fln  =  0 

has  I  different  roots  jui,  1x2,  •  •  • ,  m  occurring  wi,  712,  •  •  •,  ni  times  respectively 

(ni  +  n2  +  •  •  •  +  w;  =  n)  and  let  n'  be  the  largest  of  the  numbers  Ui,  112,  "  -,711. 

If  one  of  the  functions  t(x)  exists  such  that  for  sufficiently  large  values  of  x 

(6)  |ar«(.T)|^^J^;        r=  1,  2,  ...,7i;        s  =  0,  1,  ■  ■  •,  2n  -  1, 

then  for  the  same  values  of  x  the  equation  (4)  has  n  linearly  independent  solu- 
tions T/i,  k{x)  expressible  in  the  form 

Vi.  k(x)  =  a;^-ie-'^[l  +  e,-,  ,(x)];        i  =  1,  2,  -  -  -,  I;        k=l,2,  --  -,  m, 

where  €»,  A:(a;)  vanishes  at  infinity  to  at  least  as  high  an  order  as  that  of  Tiix), 
while  further 

yi'Mx)  =  x'-'e''^'[txi'  +  ri.fc.  s(x)];        5  =  1,  2,  . . .,  n, 

where  lim  ^  =  0." 

x=ao 

It  is  to  be  observed  that  for  the  special  case  t{x)  =  Ifx^  this  result  relates 
to  an  equation  of  the  form  (1)  (wherein  k  =  0),  and  furnishes  the  "  dominant 
terms  "  of  developments  for  the  corresponding  solutions  yi,  k{x).  Doubtless 
by  a  sufficiently  critical  examination  of  the  form  of  e,,fc(a;),  these  developments 
could  be  identified  with  asymptotic  developments  in  the  precise  sense  of  §  13. 
For  the  type  of  equation  considered,  the  result  is  seen  to  be  in  every  sense  general 
so  far  as  the  possibility  of  multiple  roots  in  (5)  is  concerned,  except  for  the 
restrictions  (6).     These  latter  when  interpreted  with  reference  to  (1)  mean  that 

(7)  ar.a=0;         r  =  1,  2,  3,  --^n;        5  =  1,  2,  3,  •  • -,  2?i' -  1 

and  hence  come  to  impose  unfortunate  restrictions.  However,  the  result  is  of 
decided  value  in  showing  that  all  further  studies  upon  the  problem  in  hand 
may  be  limited  to  those  cases  (assuming  multiple  roots  present  in  (2))  wherein 
(7)  are  not  satisfied. 

Complex  Variable 

32,  Passing  to  the  corresponding  studies  upon  (1)  when  the  independent 
variable  x  is  allowed  to  take  on  complex  values,  the  existence,  form  and  range 
of  the  asymptotic  solutions  have  been  completely  discussed  by  Birkiioff  in 
case  the  coefficients  ar{x)  (r  =  1,  2,  •  •  •,  ii)  are  developable  in  convergent  series 


70  AsYAiPTOTic  Solutions  of  Differential  Equations 

(|a;|  >  jB  =  constant  sufficiently  large)  and  under  the  assumption  that  the 
roots  of  the  characteristic  equation  (2)  are  distinct.^"  Corresponding  results 
when  multiple  roots  are  present  in  (2)  do  not  appear  to  have  been  thus  far  ob- 
tained. 

Birkhoff's  essential  result  may  be  summarized  as  follows: 
"  Representing  by  vii,  m^,  •  •  • ,  tUn  the  n  (distinct)  roots  of  (2),  let  there  be 
drawn   from   the   origin    (2=0)    the   N  =  n{n  —  \){k -\-  I)    raj'S    ("critical" 
rays)  determined  by  the  equation 

real  part  of  [{vig  —  m<)a:'=+^]  =0;        s  =^  t. 

Let  the  angles  which  these  rays  make  with  the  positive  real  axis  in  the  order 
of  their  increasing  magnitude  be  denoted  by  n,  T2,  •  •  -,  r^r  and  place  Ty+i  =  ri 
+  27r. 

Then,  corresponding  to  the  sector  Tm  =  arg  x  <  tm+i  there  exists  a  set  of 
fundamental  solutions  pr  {r  =  1,  2,  •••,  n)  of  (1)  developable  asymptotically 
in  the  forms  (2)'  where  fri^),  pr  and  Arj  continue  to  have  the  meanings  there 
indicated. 

The  set  of  solutions  satisfying  (2)'  in  the  sector  (jm,  r^+i)  differs  at  most 
by  one  solution  from  the  set  satisfying  (2)'  in  the  adjacent  sector  (r^+i,  7^+2)."^® 

Linear  Difference  Equations 
33.  If  instead  of  (1)  we  take  for  consideration  the  linear  difference  equation 

(8)     yix  +  h)  +  ai(x)y{x  +  ^  -  1)  +  a2{x)y{x  +  A  -  2)  +  •  •  •+  an{x)y(x)  =  0 

wherein  the  coefficients  oi,  02,  •  •  • ,  On  are  assumed  to  be  developable  for  large 
positive  values  of  x  either  in  convergent  series  or  asymptotically  in  the  forms 

(8)'  ar{x)^x^'^^ar,o  +  ^  +  ^+  •••];        r=  1,2,  --^n, 

1^  Trans.  Am.  Math.  Soc,  Vol.  10  (1909),  pp.  463-468.  Birkhoff  considers,  instead  of  (1), 
the  system  of  n  ordinary  linear  equations  of  the  first  order: 

(A)  ^=   t,c'iiix)yi  a  =  1,2,  ■'■,n), 

in  which  for  |a;|  >  R  -we  have 

a.-,-  (a;)  =  ai^xt  +  a.-,  y^'x'-i  +  •  •  •  +  ai/«>  +  ai,f«+i)  1+  •■■  {i,j  =  1,2,  ••■,n), 

X 

the  characteristic  equation  then  becoming 

|a,-,-  —  5,-,a|  =  0;     5,-j  =0     if     i  =t=j;     S.y  =  1     if     i  =  3. 

The  equation  (1)  may  be  transformed  into  a  sj'stem  of  the  form  {A)  by  placing  y\  =  a;"*;/, 
7/1  =  x<"~i>*y',  •••,  y„  =  x*2/'"~^>  in  which  case  we  find  q  =  k.  Thus,  whatever  appUes  to  {A) 
applies  to  (1)  as  a  special  case  with  q  =  k. 

The  important  case  in  which  the  coefficients  ar(x)  of  (1)  are  rational  polynomials  was  dis- 
cussed in  a  series  of  earlier  papers  by  Horn  whose  results  are  summarized  by  Van  Vleck  in  the 
Boston  Colloquium  Lectures  (1905),  pp.  85-92. 

1^  For  the  precise  nature  of  this  dependence,  see  Birkhoff,  I.  c,  p.  468. 


Linear  Difference  Equations  71 

k  being  zero  or  a  positive  integer,  we  have,  corresponding  to  the  first  result  cited 
in  §  31,  the  following: 

"  If  the  roots  mi,  m^,  ••-,?««  of  the  characteristic  equation 

(9)  TO"  +  ai,om"-i  + h  fln.o  =  0 

are  distinct  and  no  one  of  them  equal  to  zero,  equation  (8)  possesses  n  linearly 
independent  solutions  yi,  y^,  •  •  - ,  yn  valid  for  large  positive  values  of  x  which  are 
developable  asymptotically  in  the  forms 

(9)'  yr  -  [r(;r  +  l)fm/x<'^  Z  -^fr' ;        r  =  1,  2,  .  • . ,  n, 

j=0     X 

where  ^r.o  =  1."^^ 

In  case  (9)  has  multiple  roots,  or  a  zero  root  (an,o  =  0)  the  principal  results 
thus  far  obtained  appear  to  be  those  of  Norlund  who  employs  asymptotic 
"  faculty  series  "  and  allows  the  independent  variable  x  to  range  over  complex 
as  well  as  real  values.  Using  his  notation  and  including  for  the  sake  of  complete- 
ness the  case  of  distinct  roots,  his  results  are  as  follows  '}^ 

"  Given  the  linear  difference  equation 

k 

(10)  Z  P^{x)u{x  -  i)  =  0, 

1  =  0 

where  the  coefficients  are  faculty  series  of  the  form 

Pi{x)  =  co^')  +  — _|_-y  +  (a:+  l)(a:  +  2) 
(11) 


(0 

{x+l){x-r2){x-\-3) 


+  7— r-rT^V7^v7r-^^+  •••;       i  =  0,1,2, -••,  k 


all  of  which  converge  throughout  the  right  half  of  the  x  plane.^^  Suppose  first 
that  the  roots  ai,  a^,  az,  •  •  •,  a^  of  the  characteristic  equation 

(12)  co^o^z*^  +  Co">2^-^  + V  co^^)  =  0;        Co^"^  4=  0,        Co^^'^  +  0 

are  distinct.  Then  there  exist  k  solutions  U\,  ii2,  -  •  -,Ukoi  (10)  such  that  through- 
out the  sector  —  (7r/2)  +  e  <  arg  x  <  (7r/2)  —  €  (e  arbitrarily  small  and  >  0) 
we  have 

nQ^  ■     .      r(.T  +  1) 

(13)  ^.— «/r(.r-p,+  l)^^^^^' 

where  py  is  a  constant  and  <pj{x)  a  faculty  series  of  the  form  indicated  in  (11). 

1'  Cf.  Horn,  Journ.fiir  Malh.,  Vol.  138  (1910),  p.  159. 

18  "Kongelige  Danske  Videnskabcrncs  Selskabs  Skriftcr  "  (M6m.  de  I'Acad.  Roy.  des  Sciences 
et  des  Lettrcs  de  Denmark),  Vol.  6  (1911),  pp.  317-318.  It  would  appear  that  the  proofs  of  the 
results  here  stated  have  not  as  yet  been  published  except  in  part. 

1^  A  very  broad  class  of  series  of  the  form  (11)  have  this  property.  See  for  example  Nielson  , 
"Handbuch  der  Theorie  der  Gammafunction,"  Leipzig  (Teubner),  1906,  §  96. 


72  Asymptotic  Solutions  of  Differential  Equations 

In  case  (12)  has  multiple  roots  and  a/  is  an  n-fold  root,  Norlund  distinguishes 
two  cases: 

(1)  aj  is  at  the  same  time  an  {n  —  p)-fold  root  of  the  equations 

k 

YjCp^'h^-'  =0;        p  =  1,  2,  •  •  •,  w  -  1. 

(2)  These  conditions  are  not  fulfilled. 

In  (2)  no  asymptotic  development  exists  of  the  form  (13). 
In  (1)  there  exist  n  linearly  independent  solutions  Us{x)\  s  =  \,  2,   •••,  n 
such  that  when  —  (x/2)  +  e  <  arg  x  <  (7r/2)  —  e  we  have 

Us  ~  a/^s{x);        s  =  1,  2,  3,  •  •  •,  n, 
where 

^r  ^           f  ^      r(a:+  D        ,        ,  ,  ^        r(:r  +  1)        , 
^s(x)  =  (po{x)=rr- 777^+  ^iW 


T(x  -  ps+l)   '    "^'^  '  dp,  T{x  -  ps+l)   ' 

,         .^3"         r(a:+l) 


dp/  T{x  -  ps+1)' 

the  expressions  (po,  (pi,  •  •  • ,  <Pn  being  developments  of  the  form  (11). 

If  some  of  the  roots  of  (12)  are  zero  or  infinite,  it  is  necessary  in  order  to  obtain 
a  system  of  fundamental  solutions  to  use  a  series  of  substitutions  of  the  form 

u{x)  =  [V{x)Y^u'^''^\x)  =  T'''^{x)u^^'-\x) 

and  determine  p.r  so  that  the  difference  equation  in  w^'^'^(x)  shall  have  a  charac- 
teristic equation  containing  at  least  one  root  which  is  finite  and  different  from 
zero.  It  is  always  possible  to  determine  in  but  one  way  a  series  of  numbers 
Mi>  M2>  •  •  •  J  Mm  such  that  the  total  number  of  roots  which  are  finite  and  different 
from  zero  in  the  corresponding  characteristic  equations  thus  obtained  is  exactly 
the  order  k  of  (10).^°  If,  whenever  a  multiple  root  occurs  in  one  of  these  charac- 
teristic equations,  the  corresponding  conditions  under  (1)  are  satisfied,  then 
there  exists  a  system  of  fundamental  solutions  of  (10)  each  of  which  is  asymp- 
totically represented  within  the  sector  —  (7r/2)  +  e  <  arg  x  <  (7r/2)  —  €  by  a 
series  of  the  form 

V>''-{x)af^s{x). 

Exceptions  occur,  however,  when  some  of  the  numbers  iXr  are  not  integers,  since 
the  coefficients  in  the  above-mentioned  difference  equations  are  then  no  longer 
developable  in  faculty  series  of  the  form  (11).  For  example,  suppose  jur  =  a 
rational  fraction  j^jq.  We  may  then  put  x  =  yz,  u(x)  =  v(z)  and  derive  from 
(10)  a  difference  equation  for  v{z),  thus  demonstrating  the  existence  of  solutions 
expressible  asymptotically  in  the  forms 


r 

"  Norlund,  Ada  Math.,  Vol.  34  (1911),  p.  16 


"(i)  "'"'*•©• 


Linear  Difference  Equations  73 

Important  studies  of  (8)  when  x  is  complex  and  under  the  assumption  that 
the  roots  of  (9)  are  different  from  zero  and  distinct  and  that  the  coefficients 
ar{x)  are  rational  fractions  developable  in  the  forms  (8)'  (wherein  the  series  would 
then  converge  for  all  \x\  sufficiently  large),  have  been  made  also  by  Galbrun^^ 
and  by  Birkhoff,^^  with  the  essential  result  that  there  exists  a  system  of  funda- 
mental solutions  G(x)  =  yi,  ?/2,  ys,  •  •  • ,  yn  developable  asymptotically  in  the 
respective  forms  (9)'  throughout  the  right  half  of  the  x  plane,  and  at  the  same 
time  there  exists  a  second  system  H{x)  =  yi,  7/2,  •  •  -,  yn  of  fundamental  solutions 
developable  likewise  in  the  forms  (9)'  but  throughout  the  left  half  of  the  plane. 
Moreover,  the  elements  of  the  system  G{x)  when  considered  in  the  left  half 
plane  possess  asymptotic  developments  other  than  (9)'  whose  forms  change  as 
arg  x  passes  through  any  one  of  certain  radial  directions  ("  secondary  critical 
rays  ")  lying  in  the  second  and  third  quadrants,^^  while  similarly  the  elements 
of  H{x)  when  considered  in  the  right  half  plane  are  developable  asymptotically 
in  forms  differing  from  (9)'  and  changing  as  arg  x  passes  through  certain  radial 
directions  situated  in  the  first  and  fourth  quadrants. 

Returning  again  to  the  case  in  which  x  is  regarded  as  real  and  positive  and 
assuming  further  that  it  is  confined  to  integral  values,  we  have,  corresponding  to 
the  last  result  stated  in  §  31,  the  following  i^'^ 

"  Let  t(x)  be  one  of  the  system  of  functions  (3)'  and  put 

00 
Ti{x)  =     S    r(.Ti). 

Xl=Z+l 

Suppose  now  there  is  given  a  difference  equation 

[ao  +  aQ{x)]yix  +  n)  +  [ai  +  ai(x)]yix  +  ?i  —  1)  +  •  •  • 
(14) 

+  [«n  +  an{x)]y{x)  =  0, 
whose  characteristic  equation 

aoM"  +  aiix''-^  +  . . .  +  a„  =  0 

has  I  different  roots  )Ui,  1x2,  -  -  • ,  Hn  occurring  n\,  ni,  -  -  - ,  ni  times  respectively 
(jii -\-  712 -\-  ••■-{•  ni  =  11)  and  let  n'  be  the  largest  of  the  numbers  ni,  n2,  •  •  • ,  nu 

21  Acta  Math.,  Vol.  36  (1913),  pp.  1-68;  also  Comvt.  Rend.,  Vol.  148  (1909),  pp.  905-907. 

22  Trans.  Am.  Math.  Soc,  Vol.  12  (1911),  pp.  243-284.  As  in  his  studies  on  linear  differential 
equations  (cf.  footnote,  p.  — ),  Birkhoff  considers  a  system  of  linear  difference  equations  of  the 
first  order.  In  order  to  identify  the  forms  (9)'  with  those  occurring  in  his  results,  it  suffices  to 
observe  that 


r(x  +  1)  ~  x'+^'h-'(^co  +  ^  +  §+•••)  • 


See  for  example  Horn,  Math.  Annalen,  Vol.  53  (1900),  p.  191. 

2'  For  the  precise  statement,  see  Birkhoff,  I.  c,  p.  277-278.     See  also  p.  279,  lines  1-7. 

2*  Cf.  Love  in  Afn.  Journ.  Math.     Obtained  earlier  by  Ford  in  case  all  roots  of  the  charac- 
teristic equation  have  the  same  modulus  {Annali  di  Mat.,  Vol.  13  (1907),  p.  328). 


74  Asymptotic  Solutions  of  Differential  Equations 

If  a  function  t{x)  exists  such  that  for  sufficiently  large  values  of  x 

t(x) 
\cxrix)\^^;^^;        r  =  0,  1,  2,  ..-,  n, 

then  for  the  same  values  of  a:  the  equation  (14)  has  n  linearly  independent  solu- 
tions ?/,-,  k{x)  expressible  asymptotically  in  the  forms 

Vi,  k{x)  ~  .r^-Vi11  +  ^i.  k{x)];         i  =  1,2,  '-'J;         k  =  1,2,  ■'-,  iii, 

where  e;,  u  vanishes  at  infinity  to  at  least  as  high  an  order  as  that  of  Ti(a')." 

Summary 

34.  A  comparison  of  the  results  noted  in  §§  30-33  would  indicate  that  the 
study  of  the  asymptotic  solutions  of  either  the  differential  equation  (1)  or  the 
difference  equation  (8)  is  already  in  a  fairly  satisfactory  state  provided  the 
assumption  be  made  throughout  that  the  roots  of  the  characteristic  equation 
are  distinct,  but  much  remains  to  be  done  in  those  cases  where  multiple  roots 
are  present.  In  fact,  it  is  only  for  the  equation  (1)  of  the  special  orders  n  =  2 
or  n  =  3  that  we  find  what  could  be  described  as  a  complete  discussion,  and 
even  this  has  thus  far  been  carried  out  only  for  the  real  variable  x. 


CHAPTER  IV 

ELEMENTARY  STUDIES  ON  THE   SUMMABILITY  OF  SERIES 

35.  Introduction. — The  divergent  series 

(1)  1-  1  +  1-  1  +  1-  1+  ..• 

was  regarded  b}^  Euler^  as  having  the  sum  |  on  the  ground  that  the  expression 
1/(1  +  X)  gives  rise  by  division  to  the  series 

(2)  1-  x+  x"  -  x'  +  x^  -  x^  +  •  •  •, 
so  that  in  particular  (placing  x  =  1)  one  must  have 

(3)  i=  1-  1+  1-  1  +  1-  1+  •••. 

In  general,  the  "  sum  "  of  a  series  (convergent  or  divergent)  was  taken  to  be  the 
number  most  naturally  associated  with  it  from  the  standpoint  of  mathematical 
operations.  This  conception,  however,  naturally  led  to  inconsistency.  Thus, 
by  developing  the  expression  (1  —  .t")/(1  —  a;"")  into  the  form 

(4)  1  -  a;"  +  a;"*  -  .t"+^  +  x-""  -  •  •  • , 

and  noting  the  result  when  a:  =  1  we  obtain  for  the  series  (1)  the  sum  n/m  instead 
ofi 

The  notion  of  sum  as  thus  loosely  conceived  was  eventually  replaced  by  the 
exact  definition  of  Abel  and  Cauchy  according  to  which  the  sum  of  any  series 

(5)  ao  +  ai  +  a2  +  as  +  •  •  • 
is  taken  to  mean  the  limit 

(6)  s  =  Hm  (oo  +  ai  +  02  +  •  •  •  +  On). 

Series  for  which  this  limit  exists  were  termed  convergent,  all  others  divergent. 

Of  the  two  classes  of  series  thus  arising,  the  former  occupied  almost  exclusively 
the  attention  of  the  immediate  successors  of  Abel  and  Cauchy  and  to  such  an 
extent  that  all  divergent  series  came  to  be  regarded  as  of  questionable  value  and 
indeed  of  doubtful  significance.  It  is  a  noteworthy  fact,  however,  that  Abel 
and  Cauchy  themselves  never  ceased  to  regard  divergent  series  with  much 
interest  and  with  the  belief  that  such  series  should  by  no  means  be  banished  from 
analysis  for  the  mere  reason  that  they  fell  outside  the  pale  of  the  particular 

1  For  a  more  extended  historical  account,  see  Borel,  "  Lemons  sur  les  Series  Divergentes" 
(Paris,  1901),  Introduction. 

75 


76  Studies  on  Summability 

definition  (6).  Each  felt  on  the  other  hand  that  the  subject  presented  a  rich 
field  for  further  research. 

Only  since  the  time  of  Weierstrass  has  the  question  thus  arising — viz., 
whether  any  numerical  significance  can  properly  be  attached  to  a  divergent  series 
— been  scientifically  attacked  and  in  large  measure  answered.  The  avenue  of 
approach  has  been  chiefly  through  the  so-called  boundary-value  (Grenzwert) 
problem  in  the  theory  of  analytic  functions.^  Thus,  Frobenius^  showed  in  the 
first  place  that  if 

00 

(7)  Z  dnX"" 

n=0 

be  any  power  series  having  a  radius  of  convergence  equal  to  1,  then 

(8)  lim  2^  a„.T"  =  lim -j—. , 

ar=l— On=0  n=oo  71  -J-    i 

where  Sn  =  Oo  +  oi  +  02  +  •  •  •  +  Qn-  This  was  shown  to  be  true,  at  least, 
whenever  the  limit  indicated  on  the  right  exists.  Now,  the  first  member  of  (8) 
is  naturally  associated  with  the  series  (in  general  divergent) 

00 

(9)  Z  an, 

SO  that  it  becomes  natural  to  associate  with  the  latter  the  sum 

50  +  5i  +  52  +    •  •  •   -\-  Sn 


(10)  s  =  lim 


n-\-  1 


whenever  this  limit  exists.  Formula  (10),  regarded  as  a  general  formula  for 
defining  the  sum  of  any  given  divergent  series  (9),  finds  additional  justification 
in  the  demonstrable  fact  that  for  any  convergent  series  (9)  the  sum  as  defined  by 
either  (6)  or  (10)  is  the  same — i.  e.,  formula  (10)  is  consistent.  Moreover,  this 
selection  for  s  is  seen  to  bear  an  interesting  relation  to  the  early  statement  of 
EuLER  noted  above  respecting  the  particular  series  (1),  since,  when  applied  to  (1), 
it  gives  at  once  5  =  |. 

In  the  present  chapter  certain  general  studies  are  first  undertaken  (§§  36-40) 
upon  a  few  of  the  well-known,  standard  definitions  for  the  "  sum  "  of  a  diver- 
gent series.  The  definitions  selected  (which  include  (10)  as  a  special  case) 
are  subjected  in  turn  to  a  number  of  tests  which  it  is  believed  any  such  definition 
may  well  be  asked  to  satisfy,  and  the  results  attained  are  summarized  in  §  41. 

2  For  a  description  of  this  problem  see  Jahraus,  "  Das  Verhalten  der  Potenzreihen  auf  dem 
Konvergenzkreise  historisch-kritisch  dargestellt,"  Programm  des  Kgl.  humanist.  Gymnasiums 
Ludwigshafen  a.  Rhein  (1901),  pp.  1-56.  See  also  Knopp,  "Grenzwerte  von  Reihen  bei  der 
Annaherung  an  die  Konvergenzgrenze,"  Dissertation  (Berlin,  1907). 

8  Journ.fiir  Math.,  Vol.  89  (1880),  p.  262. 


Generalities  77 

The  underlying  principles  guiding  the  development  of  these  §§  are  stated  in 
the  Preface  and  hence  need  not  be  repeated  here. 

In  the  latter  part  of  the  chapter  the  essential  properties  of  "absolutely 
summable  "  series  are  considered  (§  42)  and  this  is  followed  by  a  few  supple- 
mentary theorems  and  remarks  on  the  theory  of  summability  in  general,  proofs 
being  suppressed  when  reference  can  be  readily  made  to  them  elsewhere. 

36.  Definitions  of  Sum. — Let  any  given  series  (convergent  or  divergent)  be 
represented  by 

(11)  f.Un 

n=0 

and  let  us  place 

n 

Sn  =   2-^  llri' 

n=0 

If  (11)  is  convergent  let  its  sum  be  indicated  by  S,  if  divergent  let  the  sum  as- 
signed to  it  by  whatever  manner  be  indicated  by  s. 

The  definitions  for  s  to  which  we  shall  confine  our  attention^  are  as  follows: 

(I)  s  =  \\m-f-(r)'y        r  =  fi^'ed  integer  ^  0     (Cesaeo),^ 

n=oo  ^n 

where 
(12) 


S  (r^  -  ,     \    rs     .1   '^'  +  ^^  .       4-  I   Kr  +  1)  •  •  •  (r  +  n  -  1) 

'^-  ni 

J.  (,)  _  (r+l)(r+2)  ■••  (r+n) 


n\ 


Under  (I)  is  thus  included  as  a  special  case  corresponding  to  r  =  1  the  definition 
(10).  The  least  value  of  r  for  which  the  second  member  of  (I)  exists  is  called  the 
degree  of  indeterminacy  of  the  series  (11). 

*  We  have  confined  the  attention  to  what  may  be  called  the  older  and  best  known  forms  of 
definition,  (I)  and  (II)  being  connected  with  the  early  studies  of  Holder  and  Cesaro  upon  the 
boundary  value  (Grenzwert)  problem  for  functions  defined  by  power  series  (see  §  35),  while  the 
remainder,  especially  (III)  and  (IV),  are  connected  with  the  independent  and  now  classical 
studies  of  Borel  upon  divergent  series.  A  form  of  definition  prominent  in  the  more  recent 
literature,  especially  in  England,  is  that  of  Riesz  {Compl.  Rend.,  July,  1909) : 

s  =  lim   X)  Wf  (  1  -  ~  )  ;         r  =  integer  ^  0. 

n=iio  u=u  \  ' *  / 

There  should  be  mentioned  also  the  following  definition  of  De  La  Vall6e  Poussin  (Bulletins  de 
la  classe  des  Sciences  de  V Academic  Royale  de  Belgique,  1908,  pp.  193-254) : 

,  -  lim   (  V  A--^   n(n  -  1)  •  • .  (n  -  fc  +  D         \ 

' - i™  T"  +  h (n  +  i)(n  +  2)...-(ir+T)  'V • 

For  a  general  study  of  possible  forms  of  definition,  see  Silverman's  Thesis  "On  the  definition  of  the 
sum  of  a  divergent  series"  in  the  scientific  publications  of  the  University  of  Missouri  for  April 
1913,  pp.  1-96.  ' 

5  Bidlctin  des  Sciences  Math.  (2),  Vol.  14  (1890),  p.  119.  Chapman  has  extended  the  defini- 
tion to  include  fractional  values  of  r  (Proc.  London  Math.  Sac,  Vol.  9  (1911),  pp.  369-409). 


(ID 

where 


Studies  ox  SuMiL\BiLiTY 
s  =  lim  Sn^''^;        r  =  fixed  integer  ^  0     (Holder),^ 

<?    (0)    _    „ 


*„">   =  ^^^  (So''^  +  ^1^°>  +    •  •  •   +  Sn^'^), 


Sn 


.  *n 


(2)    = 


71+   1 


(>•)  = 


1 


(5o"^  +  51^1)  +    •  •  •   +  5n«), 


(^o^--^)  +  51^'-"  +    •  •  •   +*n^^^0 


(HI) 


n+  1 
s  =  lim  e~''5(a)     (Borel),^ 


where  5(a)  is  defined  by  the  following  series  (assumed  convergent  for  all  values 
of  a) 

(13)  5(a)  =  i:^  a". 

(IV) 


,=0  n : 


e'"  u  (a)  da     (B  orel)  / 


where  u{a)  is  defined  by  the  following  series  (assumed  convergent  for  all  values 
of  a) 

n{a)  =^  t'^.a'^. 


(14) 


e~''Up{a)da;        p  =  fixed  integer  ^  1, 


(V) 
where 

Wp(a)  =  (wo  +  '^1  +  •  •  •  +  Vp-i)  +  {lip  +  Up+i  +  •  •  •  +  «2p-i)a 

+   {U2p  +    •  •  •    +   UZp-lW  + 

(VI)  s=  r  e-''Up{a)da,' 

Jo 

where 


(15) 


Up{a)  =  2 


2/„a' 


„=o  (np) !  ' 


p  =  fixed  integer  ^  1 . 


37.  Consistency  of  the  Above  Definitions. — It  is  at  once  to  be  assumed  that 
any  tenable  definition  of  sum  for  divergent  series  must  be  such  that  in  the  case 

«  Math.  Annalen,  Vol.  20  (1882),  pp.  535-549. 

^Cf.  "Lemons,"  p.  97. 

8Cf.  "Lemons,"  p.  98. 

s  Due  to  LeRoy.     Cf.  Annates  de  la  Faculle  des  Sciences  de  Toulouse  (2),  Vol.  2  (1902),  p.  217. 


Consistency  of  Definitions  79 

of  a  convergent  series  it  gives  s  =  S.  This  property  of  a  definition  is  called  its 
consistency}^  We  proceed  to  establish  the  consistency  of  all  the  above  definitions 
by  a  uniform  method  based  upon  the  following  general  lemma  in  the  theory  of 
limits.^^ 

Lemma. — Let  so,  Si,  s^,  •  •  • ,  Sn,  •  •  •  be  a  sequence  of  quantities  (real  or  com- 
plex) such  that  lim  Sn  =  I  and  let  ao^^^  a/^^  02^^\  •  •  •,  On^^^  •  •  •  be  a  sequence 

71=00 

of  positive  quantities  (weights)  dependent  upon  a  parameter  p  (independent 
of  n).    Also  let  it  be  supposed  that  the  expression 


Sp  = 


.Z^  Ctji       Sn 


has  a  meaning  for  every  value  of  p  in  a  given  sequence  P  of  positive  elements 
which  increase  indefinitely  to  +  oo.  If,  then,  p  be  allowed  to  increase  in- 
definitely ranging  over  the  values  in  P  we  shall  have  lim  Sp  =  I  provided  that 

p=+» 

(A)  lim  '^^^—  =  0, 

«=o 

where  m  is  any  fixed  positive  integer  (independent  of  p  and  n). 

Proof. — We  have  by  hypothesis  Sn  =  1+  e„ ;  lim  €„  =  0  and  it  suffices  to 

K  =  00 

show  that  lim  Dp  =  0,  where 


p=X) 


^  P  00  ''• 


By  writing 


n=0  n=0  n=mH-l 

and  then  placing  5„  =  /  +  fn  in  the  last  term  here  appearing  we  obtain 

m  00 

E(^n-/)a„^'^>+      Z      €„fl„(^) 

P  00  • 

n=0 

loCf.  BuoMwiCH,  "Infinite  Series"  (London,  190S),  §  100. 

"  Cf.  FouD,  American  Journ.  of  Malh.,  Vol.  32  (1910),  p.  320.  As  here  generalized,  the 
lemma  was  first-  obtained  and  applied  to  the  discussions  of  the  present  chapter  by  Meni)enh.\ll 
in  his  thesis  entitled  "  On  the  Characteristic  Properties  of  Sum-Formulaj  in  the  Theory  of  Divergent 
Series,"  University  of  Michigan,  1911. 


80 


Studies  on  Summability 


"Whence,  if  we  indicate  by  g^  a  positive  quantity  such  that  gjn^\si\;i  =  0, 
1,  2,  •  •  •,  m,  we  may  write 


\Dp\^  {gm  +  \l\) 


n=0 


ZaJ^' 


This  relation  holds  good  for  any  preassigned  value  of  p  belonging  to  P  and 
for  any  preassigned  arbitrarily  large  positive  integral  value  of  m.  The  same 
having  been  once  established,  let  us  now  choose  an  arbitrarily  small  positive 
quantity  e  and  then  take  m  so  large  that  ]  e„|  <  «;  n  =  m  +  1,  to  +  2,  •  •  •. 
We  may  then  write 

£    \en\aj^'<e\f:an''^-  JlaJ-A. 

n=m+l  L  n=0  n=m  J 

Whence, 


Dp  <  igm  +  \l\) '^ +  € 


1  - 


from  which  the  desired  result  follows  as  soon  as  we  introduce  the  hypothesis  (A). 
38.  We  may  now  easily  show  the  consistency  of  definition  (I).     For  this 
purpose  let  us  take  P  in  the  lemma  of  §  37  as  the  sequence  of  positive  integers 
0,  1,  2,  3,  •  •  •,  and  let  a„^P^  be  defined  as  follows: 

,,       r(r  +  1)  •  •  •  (r  +  2>  —  n  —  1)        . 

a„(p>  =  -^— ^— ^ — -^ f -'     when    n  <  p; 

{p  —  n)\ 

a„^p)  =  1     ivhen    n  =  p;        aj^^  =  0    when    n  >  p. 

Then  S^  =  Sp'^'^Dp^''^  where  Sp^'^  and  Dp^'^  are  given  by  (12).     Condition 
(A)  of  the  lemma  is  satisfied  since 


ZaJ^'  Zy 

iiS  Ir^  "  ll'S  (r+l)(rV2)  ■■'{r  +  p) 


n=0 


=  Hmr 

p=a)  L 


P 


+ 


rp 


r+p      {r  -\-  p){r  -{-  p  —  1) 

rp{p  —  1)  •  •  •  (p  —  m  +  1) 


+ 


(r  +  2^)ir+  p  —  I)  •  ■•  ir+  p 
Thus  we  have  the  desired  result : 

lim  Sp  =  lim  Sn  =  S, 


^1  =  0. 

-  m)  J 


p=00 


provided  the  latter  limit  exists,  i.  e.,  when  (11)  is  convergent. 


Consistency  of  Definitions  81 

The  consistency  of  (II)  follows  directly  from  that  of  (I)  if  we  make  use  of 
the  following  established  result:  "If  the  limit  s  defined  by  (II)  exists  for  a 
given  value  of  r  then  the  limit  s  defined  by  (I)  exists  for  the  same  value  of  r,  and 
conversely.  Moreover,  the  two  limits  s  are  the  same."  In  view  of  this  result 
it  appears  that  formulse  (I)  and  (II)  are  coextensive  both  in  applicability  and 
in  the  values  of  s  which  they  associate  with  a  given  series  (convergent  or  di- 
vergent).    As  the  proof  of  the  indicated  result  is  lengthy,  it  will  be  omitted  here.^^ 

To  show  the  consistency  of  (III),  let  P  be  taken  as  the  continuous  domain 
p  ^  0  and  let  aj^^  =  p^'/n !.    Then 

Sp  = -^-^ =  e-Ps{p). 

Condition  (A)  is  satisfied  since 

in    „m 

(16)  lime-^E— ,=  0. 

p=oo  n=0  ''i  • 

Thus  the  lemma  yields  the  desired  result : 

(17)  lim  e~^s{p)  =  lim  e'^sia)  =  lim  Sn  =  S. 

p—oo  a=oo  n=ao 

In  considering  the  consistency  of  (IV),  we  first  note  that  when  (11)  is  con- 
vergent, lim  Un  =  0.     Whence,  if  we  apply  the  lemma  of  §  37  with  s„  =  w„  and 

n=oo 

fln^^^  =  p"/n!,  noting  also  relation  (16),  we  obtain 

(18)  lim  e'^uip)  =  lim  e"'u{a)  =  lim  Un  =  0, 

p=ai  a=co  n=« 

where  u(a)  has  the  meaning  given  in  (14). 

Now,  from  equation  (17)  together  with  [e~''s{a)]a=o  =  uo,  we  may  write 

r  d 


r     d 

S  —  Uo  —    I     ^[e~°-s{a)]da. 


But 

^[e-"5(a)]  =  e-V(a)-5(a)], 
where 

-^s{a)  =  s'{a)  =  5i  +  52o:  +  *3  .yj  +  •  •  • . 

Whence,  if  we  note  that 

d  a^ 

^u{oc)  =  u'{a)  =  s'ia)  —  s(a)  =  Ui  +  i^a  +  U3^^-{-  ■■■, 

"See  Ford,  I.  c,  pp.  315-326.  Also  Schnee,  Malh.  Annalcn,  Vol.  67  (1909),  pp.  110-125. 
In  view  of  this  result  we  shall  omit  the  detailed  discussion  of  (II)  throughout  the  present  chapter, 
all  statements  respecting  it  being  identical  with  those  obtained  for  (I). 


82  Studies  on  Summability 

we  have 

/»« 

(19)  S  -  uo=         e-''u'{a)da. 

Jo 

Whence  also,  upon  integrating  by  parts, 

S  -  Wo  =  U""  /    u'(a)da       +1     e"'      j    u'{a)da  Ida 

e-^iiiia)  -  uo]   \    +1     e-^iuia)  —  Uo}da. 

Introducing  (18)  together  with 

11 0  =    I     e~'^iioda 
Jo 
we  reach  the  desired  relation : 

(20)  I     e-''u(a)doi  =  S. 

Jo 

Definition  (V)  is  at  once  seen  to  be  consistent,  for  when  (11)  converges  to  S 
so  also  does  the  series 

{llO  +  Wi  +    •  •  •   +  Up-l)  +   («p  +  Up+l  +    •  •  •   +  «2p-l) 

H-   {U2p  +  W2H-1  +    •  •  *   +  UZp-l)  +    •  •  •, 

and  by  applying  (IV)  to  this  series  we  obtain  the  desired  result: 


f 


e  °'iLk{ot)da  =  S. 


Likewise,  the  consistency  of  (VI)  may  be  shown  by  use  of  (20)  for  it  is  merely 
the  application  of  this  equation  to  the  series 

«o  +  0  +  0  + h  0  +  wi  +0+0+.--+W2+0+-.-, 

wherein  p  —  1  zeros  are  inserted  between  each  term  and  the  preceding  term 
in  (11). 

39.  The  Boundary  Value  Condition. — It  is  well  known  that  two  definitions 
of  sum,  both  "  consistent  "  (§  37),  do  not  necessarily  give  the  same  sum  to  a 
given  divergent  series.  In  other  words,  consistency  alone  is  not  an  adequate 
principle  upon  which  to  base  a  scientific  theory  of  summation  because  it  does 
not  insure  uniqueness  of  sum.^^     A  theory  free  from  this  objection  may  be 

"  See  remarks  in  Preface.  It  would  appear  that  many  of  the  formulae  for  sum  suggested 
within  recent  years  have  been  obtained  from  considerations  quite  regardless  of  the  question  of 
uniqueness. 


The  Boundary  Value  Condition  83 

attained  if  (having  demanded  consistency)  we  confine  the  attention  to  those 
series  (11)  for  which  the  corresponding  power  series^^ 

(21)  f(x)^f:2lr.X- 

n=0 

has  a  radius  of  convergence  equal  to  1  and  then  agree  to  retain  those  definitions 

of  sum  for  which 

(B)  s  =   lim  fix). 

a:=l-0 

This  procedure  is  in  Hne  with  the  historical  genesis  of  the  theory  of  summability 
and  allows  the  theory  a  well-defined  usefulness  in  the  study  of  analytic  functions.^^ 
Indeed,  if  a  general,  self-consistent  theory  is  to  be  formulated,  it  would  seem  that 
it  should  contain  (B),  or  an  equivalent  condition,  though  such  a  condition 
evidently  tends  to  hmit  the  immediate  range  of  appHcability  of  the  theory  to  a 
particular  class  of  series  (11)  (cf.  Preface). 

Having  assumed,  then,  that  the  series  (11)  is  such  that  the  power  series  (21) 
has  a  radius  of  convergence  equal  to  1,  we  shall  undertake  to  determine  in  the 
present  §  those  definitions  of  sum  which  satisfy  (B).  Definitions  having  this 
property  we  shall  speak  of  as  satisfying  the  boundary  value  condition. 

We  begin  by  showing  that  definition  (I)  satisfies  (B),  i.  e., 

(22)  lim  flunx^  =  lim  Sn^^yDJ^\ 

x=l—On=0  n=oo 

whenever  the  latter  limit  exists.  This  may  be  done  as  follows  by  the  aid  of  the 
lemma  of  §  37. 

Let  the  Sn  of  the  lemma  be  taken  as  SJ''^IDJ''\  Then  place  x  =  1  —  1/p 
so  that  as  x  ranges  from  a  to  1  (0  <  a  <  .t)  the  quantity  p  ranges  from  1/(1  —  a) 
to  +  oo;  also  take  aj^^  =  DJ'\1  -  l/^;)".  The  expression  Sp  of  the  lemma 
then  becomes 

n=0  \  PJ  ji=0 

'■•  It  is  to  be  observed  that  tliis  scries  is  formed  by  supplying  the  successive  powers  of  x  into 
(11)  beginning  with  x°,  thus  excluding,  for  example,  the  series  (4)  in  connection  with  the  study 
of  (1).  This  choice  oi  fix),  though  arbitrary,  is  evidently  the  most  natural  and  the  one  most 
likely  to  result  in  a  theory  of  summability  having  useful  supplemental  relations  to  the  boundary- 
value  problem. 

15  Some  sum-formulae,  such  as  (IV),  §36,  not  only  satisfy  (B)  when  applied  to  series  (11) 
for  which  (21)  has  a  radius  of  convergence  equal  to  1,  but  they  have  the  further  property  that 
they  preserve  a  meaning  in  certain  regions  in  which  |x|  >  1  and  in  these  regions  furnish  the 
analytical  continuation  of  (21)  (cf.  §  44). 


84  Studies  on  Summability 

so  that 

00 

lim  Sp  =   lim  X  w„a:". 

p=«)  2=1—0  n=0 

Let  us  now  confine  ourselves,  as  may  be  done  without  loss  of  generality,  to 
values  of  p  pertaining  to  the  sequence  P  =  1,  2,  3,  •  •  •.  Condition  (A)  of  the 
lemma  is  now  satisfied,  since 

which  expression  is  evidently  equal  to  zero  since  the  denominator  has  a  meaning 
for  all  2^  >  0  but  becomes  infinite  with  p,  while  the  numerator  remains  finite  as 

p   =    00. 

Applying  the  lemma,  we  may  therefore  write  (22)  as  desired. 

We  turn  next  to  definition  (III)  and  shall  show  that  (B)  is  again  satisfied,  i.  e., 

00 

(23)  lim  Xlw„a:"  =  lim  e~''s{a), 

Z=V—0  n=0  a=oo 

whenever  the  latter  expression  has  a  meaning. 

For  this  purpose  we  first  note  that  for  any  series  (11)  (convergent  or  di- 
vergent) for  which  the  second  member  of  (22)  exists  we  have  in  the  notation  of 
§36 

(24)  ^[^~"*(«)]  =  e-'^Wifii)  -  s(a)]  =  e-^u'ia);         [e-«5(a)]„=o  =  ^o 

and  hence 

^  [e-''s(a)]da  =  Wo  +    I     e~"w'(a)c^a. 

Conversely,  it  appears  from  the  same  relations  (24)  that  for  any  series  (11) 
for  which  the  last  member  of  (25)  exists,  the  second  member  of  (23)  exists  also 
and  we  have  relation  (25). 

This  premised,  let  us  return  to  the  series  (21).     Since  this  series  is  convergent 
when    I  a:  1  <  1  it  follows  from  the  consistency  of  definition  (III)  that  when 
0<  a;<  1 

^UnX""  =  Hm  e~X(«)> 

ji=0  a=oo 

where  s^(a)  represents  the  function  s{a)  corresponding  to  the  series  (21) .    Whence, 
upon  applying  (25),  we  have  also 


The  Boundary  Value  Condition  85 

(26)  X]«n.^■"  =  Uo-\-  I     e'^u' {ax)da;         u'{ax)  =  ■^u{ax). 

Assuming  for  the  moment  that  the  integral  here  appearing  converges  uniformly 
for  all  values  of  x  in  the  interval  a<a:<l;a>Owe  now  have,  using  (25), 

00  /»oo 

hm  ^UnX'"'  =  2<o  +  I     e"'u'{a)da  =  lim  e'^sia), 

a;— 1—0  «=0  t/0  a=oo 

thus  reaching  the  desired  relation  (23). 

That  the  integral  in  (26)  converges  uniformly  for  values  of  x  in  the  interval 
a  <  a:  <  1  may  be  established  as  follows:  Place  ax  =  y  and  subsequently  replace 
X  by  1/(1  +  6).     The  integral  under  consideration  thus  takes  the  form 

(27)  (1  +  0)    r   e-'ye-Hi'{y)dy, 

so  that  it  now  suffices  to  show  that  (27)  converges  uniformly  for  all  values  of  $ 
in  the  interval  0  <  6  <  b;  b  =  {1  —  a)fa. 
Now,  the  integral 


/»oo 

(28)  e-^u'{y)dy 

Jo 


converges,  as  appears  from  (25),  when  we  make  use  of  our  hypothesis  that  the 
second  member  of  (23)  exists.  Moreover,  the  expression  e~^^  is  positive  and 
steadily  decreasing  as  y  increases  and  it  becomes  equal  to  1  for  all  values  of  6 
when  y  =  0.  We  have  therefore  but  to  apply  Abel's  test^^  for  the  uniform 
convergence  of  definite  integrals  to  reach  the  desired  result  concerning  (27). 
We  proceed  to  show  that  definition  (IV)  also  satisfies  condition  (B),  i.  e., 


(29) 


00  /^OO 

lim  X^  UnX"^  =    I     e~'^u{a)da, 

a;=l— 0  ra=0  Jq 


whenever  the  latter  expression  has  a  meaning. 

From  the  consistency  of  (IV)  we  have  in  the  first  place 

00  /»oo 

lim  zlunX^  =   hm    I     e~''u{ax)da, 

J-— 1— On=0  x=l— Ot/Q 

so  that  it  suffices  for  our  purpose  to  prove  that 

r»ao  pan 

(30)  lim    I     c~^ii{ax)da  =    1     c~''u{a)da. 

18  Cf.  Bromwich,  I.  c,  §  171  (2). 


86  Studies  on  Summability 

Placing  ax  =  y  and  subsequently  replacing  x  by  1/(1  +  ^)/^  the  integral 
in  the  first  member  of  (30)  takes  the  form 


(1  +  0)    1     e-%-^u{y)dy 
Jo 


and  we  may  now  show  by  applying  Abel's  test,  as  in  the  discussion  of  (III),  that 
this  integral  converges  uniformly  for  all  values  of  6  in  the  interval  0  <  6  <  b; 
6  >  0,  with  which  the  proof  of  (29)  becomes  complete. 

Definition  (V)  does  not  in  general  satisfy  condition  (B),  as  appears  from  an 
example.     Thus,  let  the  series  (21)  be 

and  take  p  =  2.  Then  Up{a)  =  0  and  hence  s,  as  given  by  (V),  is  equal  to 
zero.     But, 

lim  E  (-  1)M-  =  Y^-r  =  h 

a-=l-0  n=0  ■>-      I       ■•• 

That  definition  (VI)  satisfies  condition  (B)  may  be  readily  inferred  from 
reasoning  similar  to  that  followed  in  connection  with  (IV).  Thus,  from  the 
consistency  of  the  definition  we  have 

00  /»oo 

<31)  limX!wna:"=   lim  e-''Up,xici)da, 

1=1— On=0  x—l—oJo 

■where 

Upon  placing  x  =  2"  the  second  member  of  (31)  takes  the  form 

/^oo 

<32)  lim    1     e-'Upiazjda, 

»=l-0»/0 

w^here  Up  is  defined  by  (15).  Now  place  az  =  y  and  subsequently  replace  z  by 
1/(1  +  6).     Expression  (32)  then  takes  the  form 

roo 
e-ye-'"Up(y)dy. 
.   - 

Since  the  integral 

/»« 

e-yUp{y)dy 


f 


has  a  meaning  by  hypothesis,  we  may  show  by  means  of  Abel's  test,  as  in  con- 
nection with  (IV),  that  the  integral  in  (33)  is  uniformly  convergent  for  all  values 
of  6  in  the  interval  0  <  6  <  b,  with  which  the  proof  is  at  once  completed. 
I'Cf.  Bromwich,  I.  c,  p.  121. 


Fundamental  Operations  87 

40.  Fundamental  Operations. — Besides  being  consistent  and  satisfying  the 
boundary  value  condition  (i?)/^  it  is  evidently  desirable  that  the  sum  assigned 
to  a  numerical  divergent  series  (11)  shall,  at  least  so  far  as  possible,  be  one  for 
which  the  usual  operations  applicable  to  convergent  series  are  preserved.  The 
operations  of  this  type  which  we  shall  consider  are  the  following: 

(C)  If  s  represents  the  sum  of  the  divergent  series  (11)  by  a  given  definition, 
then  the  series 

(34)  z2un;        k  =  positive  integer 

n=k 

shall  have  a  sum  5^^^  by  the  same  definition  such  that 

(35)  5(^>  =  5  -  (wo  +  ^1  +  •  •  •  +  w,_i). 

Conversely,  if  the  series  (34)  has  a  sum  s^^^  by  a  given  definition  then  the  series 
(11)  shall  likewise  have  a  sum  s  by  the  same  definition  and  relation  (35)  shall 
exist. 

{D)  If  with  a  given  definition  of  sum,  the  two  divergent  series: 


(36)  E  Un,  Z  'Vn 

»i=0  n=0 

have  respectively  the  sums  Si,  S2,  then  the  series 

Zl  (Un  ±  Vn) 


w-0 


shall  possess  by  the  same  definition  the  sum  Si  ±  52. 
(E)  With  the  hypotheses  stated  in  (D)  the  series 

(37)  tw., 

n=0 

where 

shall  have  the  sum  *i*2  (at  least  after  certain  additional  conditions  have  been 
placed  upon  w„  and  v„  analogous  to  those  imposed  when  two  convergent  series  are 
multiplied  together). 

We  begin  by  showing  that  definition  (I)  satisfies  condition  (C).  For  this  it 
evidently  suffices  to  suppose  Jc  =  1,  since  a  repetition  of  the  reasoning  leads 
from  this  to  the  general  result  (35). 

18  For  reasons  stated  at  the  beginning  of  §  39  we  shall  continue  throughout  the  present  §  to 
regard  the  given  series  (11)  as  belonging  to  the  class  for  which  the  corresponding  power  series 
(21)  has  a  radius  of  convergence  equal  to  1.  This  hypothesis,  however,  plays  no  part  in  the 
deductions  about  to  be  made. 


88  Studies  on  Summability 

Placing 

[Sn=   Uq  +  Wi  +    •  •  •    +  W„, 
[  (Tn   =    Wi  +  «2  +    •  •  •    +  lin+l, 

,   rjr  +  1)  ,  ,   r(r  +  1)  •  •  •  (r  +  n  -  1) 

iSn^^''   =   5„  +  r5„_i  H 2] *"-2  +    •  •  •   +  ~l  -^0' 

n  (.)  _  (r+l)(r+2)  •••  (r+n) 
^"      ~  n! 

we  are  to  show,  then,  that  if  the  hmit 

51  =  Urn  ,S/'-^/Dn^^^ 

n=QO 

exists  so  also  does  the  limit 

52  =  Km  iSj^^Dn^'-^ 

n=ao 

and  that  Si  =  uo  +  52,  with  the  corresponding  converse  statement. 
Since  Sn+i  =  uo  +  cr„  we  have 

r(r+l)  ,   r(r+l)--(r+n-l) 

where 

1    I      _L  ^(^  +  1)   -  -   r(r+l)  •••  (r+7^-1)  _  ^  (,) 


Whence, 


The  desired  result  (both  direct  and  converse)  now  follows  upon  noting  that 

As  regards  definition  (III),  it  appears  from  an  example  that  this  does  not 
always  satisfy  (C).  Thus,  consider  the  special  series  (11)  for  which  Uq,  Ui,  u^,  •  •  • 
are  so  determined  that 

sin  (e")  =  Uo  +  {uo  +  ni)a  +  {uq  +  Wi  +  W2)  2]+  * "  '• 
For  this  series  we  have 
s  =  lim  e""  sin  (e")  =  0;        5^^^  =  lim  e-"    ;psin  (e")  —  i/o^"    =  lim  cos  (e")  — ^0, 

a=«  a=oo  Loo;  J  „=«! 

SO  that  although  s  exists,  the  same  is  not  true  of  s'^^\ 


Fundamental  Operations  89 

In  this  connection  we  may,  however,  estabUsh  the  following  noteworthy 
result : 

"  If  the  series 

00 

(39)  Ew„;        p  =  0,  1,  2,  3,  .••,^• 

are  each  summable  by  (III)  to  the  respective  values  s,  s^'^^,  s'-'^\  ■  •  •,  s''''\  then 
relation  (35)  is  satisfied." 

In  fact,  with  Sn  and  (Xn  defined  as  in  (38)  and  with 

2 

s{a)  =  5o  +  5iq:  +  52  ^  +  •  •  • , 
<j(a)  =  (To  +  o-io!  +  (r2-^,+  •  •  • , 


we  have,  since  Sn+i  =  wo  +  ctu, 


a^  r  q;2 

s{a)  =  Wo  +  (uo  +  0-0)0;  +  (wo  +  o-i)  75-,  +  •  •  •  =  Uoe"  +     (7o«  +  ci  ^,  + 


a' 


Or,  since  (r„  =  Cn+i  —  w„+2, 

s(a)  =  Moe"  +     ((7i  —  V2)a  +  (0-2  —  W3)  ^  + 

Whence  also 

e-"5(Q!)  =  Uo  +  e~"(T(Q;)  —  e-^u'ia), 

where  u{a)  is  determined  from  (14).  It  therefore  remains  but  to  show  that 
lim  e~'^u'{a)  =  0,  in  order  to  prove  the  indicated  statement  for  the  case  in  which 
k=  1. 

Now,  having  assumed  that  both  lim  e~'^s(a)  and  lim  e~V(a)  exist,  it  follows 
from  the  last  equation  that  lim  e~'^2i'(a)  exists  also.     Moreover,  we  have  (cf.  (25)) 


(40) 


/»00 

lim  e~"-s(a)  =  Wo  +    I     e'''u'(a)da 

a=oo  t/o 


SO  that  the  only  possible  value  of  lim  e~"-u'{a)  is  zero.^^ 

Repetition  of  the  reasoning  now  leads  to  the  more  general  result  as  stated 
above. 

Turning  to  definition  (IV),  it  again  appears  that  a  series  (11)  which  is  sum- 
mable by  this  definition  may  not  satisfy  (C),  but  that  the  following  result  may 
be  established:-'^ 

"  If  the  series  (39)  are  summable  by  (IV)  to  the  respective  values  s,  5^^\ 
5(2)^  •  •  •,  s^''^,  then  relation  (35)  is  satisfied." 

*^  It  may  be  obsei'ved  that  the  existence  of  the  integral  in  (40)  does  not  suffice  to  establish 
the  equation  lim  e~"u'(a)  =  0  (cf.  Bromwich,  I.  c,  p.  278). 

2°  Cf .  Hardy,  "Researches  in  the  Theory  of  Divergent  Series,  etc.,"  Quarterly  Journ.  oj 
Math.,  Vol.  35  (1904),  p.  30. 


90  Studies  on  Summa.bility 

In  order  to  see  the  truth  of  this  statement  for  the  case  in  which  ^  =  1  we 
first  note  that  by  an  integration  by  parts  we  obtain 


(41) 


s  =    I     e~''u{a)da  =     —  e-^u{a) 

t/O  L  Ja=0 


+    1     e-''u'ia)da  =     e'^uia)  +  s^^\ 

Jo  L  Ja=0 


From  this  relation  combined  with  the  assumed  existence  of  s  and  s^^'>  it  follows 
that  lim  e~'^'u{a)  =  0  so  that  we  have  as  desired  5^^^  =  s  —  Uq.  In  order  to 
prove  the  more  general  case  we  have  evidently  but  to  repeat  the  same  reasoning 
k  times. 

Definition  (V)  does  not  always  satisfy  condition  (C)  since,  as  we  have  just 
shown,  it  does  not  do  so  for  the  special  case  in  which  p  =  1.  Likewise,  the 
same  is  true  of  definition  (VI)  (which  reduces  to  (IV)  when  p  =  1),  but  we  here 
have  an  alternative  result  similar  to  that  indicated  above. 

We  turn  then  to  condition  (D).  This  is  evidently  satisfied  by  two  series 
summable  by  any  one  of  the  definitions  of  §  36  and,  therefore,  needs  no  further 
comment. 

As  regards  condition  (E),  it  is  obviously  necessary  to  impose  further  condi- 
tions than  that  of  the  mere  summability  of  the  two  series  (36)  in  order  that  (E) 
be  satisfied,  at  least  in  generab  since  even  in  the  case  of  two  convergent  series 
such  supplementary  conditions  are  required.  We  here  have,  however,  the 
following  noteworthy  result  of  Cesaro  relative  to  series  (36)  summable  by  (I) : 

"  The  product  series  (37)  of  two  series  (36)  whose  degrees  of  indeterminacy 
(§  36)  are  respectively  r  and  s  is  summable  and  has  a  degree  of  indeterminacy  no 
greater  than  r  +  5  +  1."^^ 

Conditions  under  which  condition  (E)  will  be  satisfied  by  definition  (IV)  will 
be  considered  in  §  44. 

41.  Summary  of  Results. — The  principal  results  of  §§  35-40  may  be  sum- 
marized into  the  following  statement: 

Let 

(42)  Zun 

be  any  divergent  series  such  that  the  corresponding  power  series 

00 

"  We  omit  the  proof  of  this  well-known  result.  The  same  may  be  supplied  from  Bromwich, 
I.  c,  §  125.  For  Cesaro's  original  proof,  see  Bulletin  des  Sciences  Math.,  Vol.  14  (1890),  pp.  118, 
etc. 


Absolutely  Summable  Series 


91 


has  a  radius  of  convergence  equal  to  1.     Also,  let  (I),  (II),  (III),  (IV),  (V)  and  (VI) 
represent  the  six  definitions  for  sum  indicated  in  §  36. 

//,  then,  we  represent  by  (A)  the  condition  of  consistency  (§  37),  by  (B)  the 
boundary  value  condition  (§  39)  and  by  (C),  {D)  and  (E)  the  conditions  of  §  40 
carried  over  from  the  theory  of  convergent  series,  the  relation  of  the  various  definitions 
to  these  conditions  appears  in  the  following  table  wherein  the  *  when  placed  in  any 
square  indicates  that  the  corresponding  definition  and  condition  are  compatible: 


I 

II 

III 

IV 

V 

VI 

A 

* 

* 

* 

* 

* 

* 

B 

* 

* 

* 

* 

^ 

C 

* 

* 

D 

* 

* 

* 

* 

* 

* 

E 

Fig.  5 

Moreover,  the  squares  corresponding  to  (III,  C),  (IV,  C)  and  (VI,  C)  may  also 
receive  the  *providedlpne  substitutes  for  (C)  the  following  slightly  more  restrictive 
condition : 

{Cy  If  the  series 

00 

X)w„;        p  =  0,  1,  2,  3,  •  ••,  A; 

n=p 

are  each  summable  in  accordance  ivith  a  given  definition  of  sum  to  the  respective  values 

s,  5(»,  s^'\  s^'\   •••,  s^^) 

^(fc)   =   s  —   (^u^ -^  u^ -\-    .  .  .   -j-  Uk_i). 


then 


42.  Absolutely  Summable  Series. — A  noteworthy  class  of  divergent  series  (11) 
for  which  conditions  (A),  {B),  (C),  (D),  (E),  of  §  41  are  all  satisfied  when  we  adopt 
the  definition  (IV)  of  sum,  has  been  pointed  out  by  Borel  and  made  the  object 
of  especial  study  throughout  his  investigations."  Such  series  are  called  abso- 
lutely summable  and  are  defined  from  the  fact  that  not  only  the  integral 

/^oo 

(43)  s  =    I     e-''u{a)da 

Jo 

is  supposed  to  exist,  but  also  each  of  the  integrals 


e-''\u^^\a)\da;        p  =  0,  1,2,3, 


"  Cf.  "Legons,"  Chapter  III. 


92  Studies  on  Summability 

wherein  u^^\a)  denotes  the  pth.  derivative  of  the  (integral)  function  ii{a)  (cf. 

(14)). 

Absolutely  summable  series,  as  thus  defined,  being  but  special  series  summable 
by  definition  (IV),  at  once  satisfy  conditions  (A),  (B)  and  (D),  as  shown  in 
earlier  §§.  It  therefore  remains  but  to  consider  such  series  with  reference  to 
conditions  (C)  and  (E). 

Now,  if  the  series  (11)  is  absolutely  summable,  it  follows  from  definition  that 
both  s  and  s^''^  exist.  Whence,  by  the  results  obtained  in  §  40,  we  have  relation 
(35).  In  order  to  complete  the  proof  that  (C)  is  satisfied,  we  must  now  show 
that  if  the  series  (34)  is  absolutely  summable,  so  also  is  (11)  and  that  with  s  and 
5^*^  defined  as  before,  relation  (35)  exists.  For  this  let  us  first  consider  the  case 
in  which  k  =  1. 

Place23 


(p{x)  =    I     \u'{t)\dt^\   1    u'{t)dt 


We  thus  have 

(p{x)  ^  \u{x)  —  Wo  I 
and  hence 

|w(.r)|^  (p{x)  +  |wo|, 
so  that  the  integral 

I     e~''\u{x)\dx 
Jo 

must  converge  whenever  the  same  is  true  of  the  integral 

(44)  I     e-'<p{x)dx. 

Jo 
Now,  by  identity 

I     e-^(p(x)dx  =  -  e-^<p{X)  +   I     e-='ip'{x)dx 
Jo  Jo 

and  consequently,  because  (p{X)  and  <p'{x)  are  both  positive, 

nX  fX  r^xi 

I     e~''(p{x)dx  <    I     e~''(p'{x)dx  <    I     e~''(p'(x)dx. 
Jo  Jo  Jo 

Thus  the  integrals  (40)  and  (41)  exist.     Upon  again  applying  the  results  obtained 
in  §  40,  the  desired  conclusion  now  follows  for  the  case  in  which  k  =  1. 

A  repetition  of  the  reasoning  evidently  leads  to  the  more  general  result. 

We  proceed,  then,  to  show  that  absolutely  summable  series  satisfy  condition 

(E).24 

"  Cf.  Bromwich,  I.  c,  §  106. 

^*  The  proof  which  follows  is  essentially  that  given  by  Bromwich  {I.  c,  §  106). 


Absolutely  Summable  Series 


93 


In  the  first  place,  we  may  write  (see  definitions  of  si  and  52  in  (36),  and  note 
(43)) 

S1S2  =   lim  ffe~^'^^u{x)v{y)dxdy, 

in  which  the  double  integral  appearing  in  the  second  member  is  understood  to 
be  extended  over  the  square  OABC  of  side  X  situated  as  in  the  following  figure: 


y 

c' 

X              B' 

\ 

\ 

\ 

\ 

\ 

\ 

c 

N 

B 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

Q 

^ 

1              A! 

Fig.  6 


In  fact,  we  have 
lim   I    I  e~'-''~^"^u(x)v(y)dxdy  =  lim  I      1    e~^''^'>u{x)v{y)dxdy 

Ur»A  /»\  "1  /^oo  /»« 

e~'^u{x)dx  I    e~H{y)dy     =    I     e~''u{x)dx  I     e~yv{y)dy  =  SiSi. 

Now,  in  case  u{x)  and  v{y)  are  always  positive  the  indicated  double  integral 
when  extended  over  the  triangle  OA'C  has  a  value  lying  between  the  corre- 
sponding integrals  taken  over  the  squares  OABC,  OA'B'C ,  and  since  the  latter 
each  approach  the  limit  Sis^  as  X  =  co,  the  integral  over  the  same  triangle  will 
also  approach  the  limit  S1S2.  On  the  other  hand,  if  u{x)  and  v{y)  are  not  always 
positive,  the  absolute  value  of  the  difference  between  the  integrals  over  OABC 
and  OA'B'C  may  be  made  arbitrarily  small  by  taking  X  sufficiently  large,  as  we 
shall  show  presently,  thus  again  rendering  the  integral  over  the  triangle  OA'C 
equal  in  the  limit  to  s\Si. 

In  order  to  show  this,  let  us  represent  by  I{S)  the  integral  in  question  when 
extended  over  any  given  area  S.  Also  let  G{S)  be  the  corresponding  integral 
when  the  absolute  value  of  the  integrand  is  used.     We  then  have 


(45) 


\I{OABC)  -  1(0 A' C) I  =  \IiCBC')  +  I(AA'B) 


\I(CBC)\+\IiAA'B)  I  <  GiABCCB'A'A). 


Since  ABCCB'A'A  =  OA'B'C  -  OABC  and  since  the  integrand  of  G(S) 
is  always  positive,  the  last  member  of  (45)  may  be  written  in  the  form 


(46) 


/^2A  /^2A  /»A  /^A 

I     e~^\u{x)\dx  I     e~"\v{y)\dy  —    I    e~^\u(x)\dx  I    e~"\viy)\dy. 
Jo  Jo  Jo  Jo 


94  Studies  on  Summability 

Moreover,  since  the  series  (36)  are  by  hypothesis  absolutely  summable,  each  of 
the  iterated  integrals  in  (46)  approaches  the  same  limit  when  X  =  co,  so  that 
the  expression  (46)  itself  approaches  the  limit  zero. 
We  may  therefore  in  all  cases  write 

(47)  S1S2  =  Yimj  J  e~''^'^^hi(x)v{y)dxdy, 

A=oo 

where  the  integration  is  performed  over  the  right  triangle  OA'C,  the  length  of 
whose  side  is  2X. 

This  result  being  premised,  let  us  now  introduce  into  the  second  member  of 
(47)  the  new  variables  ^,  rj  defined  as  follows:  x  -\-  y  =  ^,y  =  ^rj  or  x  =  ^(1  —  77), 

y  =  ^^• 

We  then  have^^ 

dydx  =  d^drj  =  ^d^drj, 


dx 

dx 

a| 

dv 

dy 

dy 

d^ 

d-q 

so  that  the  integral  in  question  becomes 

(48)  e-^^d^       u{^{l  -  v)mv)dr}. 

Concerning  the  limits  of  integration  here,  we  wish  to  integrate  over  that 
area  in  the  ^,  rj  plane  which  corresponds  to  the  area  of  the  triangle  OA'C  in  the 
X,  y  plane.  Now,  the  three  sides  of  the  triangle  are  respectively  x  =  0,  y  =  0 
and  X  -\-  y  =  2X,  and  our  first  problem  is  to  determine  what  these  bounding  lines 
become  in  the  ^,  77  plane,  it  being  understood  as  indicated  above,  that  the 
equations  of  transformation  are  .T  =  ^(1  —  r]),y  —  ^V'  Evidently,  corresponding 
to  a:  =  0  we  have  the  two  lines  ^  =  0,  rj  =  1,  while  corresponding  to  y  =  0,  we 
have  the  two  lines  ^  =  0,  rj  =  0,  and  corresponding  to  x  -]-  y  =  2\  we  have 
the  one  line  ^  =  2X.  The  area  bounded  by  these  four  lines  is  that  of  the  rectangle 
whose  vertices  (in  the  ^,  77  plane)  are  (0,  0),  (2X,  0),  (2X,  1),  (0,  1).  Whence,  the 
limits  of  integration  are  as  indicated  in  (48). 

The  series  for  u(x)  and  v{y),  being  power  series  are  absolutely  convergent. 
Hence,  by  the  rule  for  the  multiplication  of  two  such  series,  it  appears  that  the 
expression  u[^{l  —  'r])]v{^r])  may  be  expanded  into  a  series  whose  nth.  term  is 

(49)  rE^r^^r,— (l-T?)". 

r=o^!(^  —  r) ! 

Moreover,  this  series  will  be  uniformly  convergent  as  regards  tj  throughout  the 
"  See,  for  example,  Gotjrsat,  "Cours  d'Analyse,"  Vol.  I  (1902),  p.  298 


Absolutely  Summable  Series  95 

interval  0  <  77  <  1  since  for  all  such  77  values  the  term  (49)  is  less  in  absolute  value 
than 

^  .t^r!(n-r)I 

and  this  expression  is  the  nth  term  of  the  (convergent)  product  series  obtained 
by  multiplying  together  the  (convergent)  series 

**    I  -)/  I  ^    1 11  I 

»=o  nl  n=o  nl 

The  integration  with  respect  to  77  in  (48)  may  therefore  be  performed  term  by 
term  upon  the  series  whose  nth.  term  is  (49),  thus  giving 

fum  -  v)mv)dv  =  i:^"E  ..y'""!.,  fr-^a  -  vYdv. 

Jo  71=0  r-Qi-V''  ')-Jo 

But 

r    n-m         ^r,        rl(n-r)l 


Thus  we  have 


/»!  00 

1     w[^(l  -  v)H^v)dv  =  Z)  Wn 

t/0  n=0 


(n+1)!' 


where  Wn  has  the  meaning  used  in  condition  (E)  (§  40). 
The  integral  (48)  thus  becomes 


/•2A 

Jo 


where 

and  accordingly  we  have  the  equation 

(50)  5i52=    re-nva)d^. 

Jo 

The  second  member  of  this  equation  is  seen  to  be  the  sum  of  the  series 

(51)  O  +  W0+W1-] , 

so  that  our  final  result  will  now  follow  as  soon  as  we  show  that  under  the  existing 
hypotheses  the  series 

(52)  Wo  +  ici  +  «'2  +  •  •  • 

is  summable  by  definition  (IV). 


96  Studies  on  Suivimabilitt 

We  may  in  fact  show  that  the  series  (52)  is  absolutely  summable.  Moreover, 
since  we  have  shown  that  absolutely  summable  series  satisfy  condition  (C), 
it  will  here  suffice  to  show  that  the  series  (51)  is  absolutely  summable — i.  e.,  that 
the  integrals 

re-f|irW(?)M^;        A;  =0,1,2,3,  ••• 

converge.  The  proof  of  this  presents  no  difficulties  and  will  therefore  be  omitted.^^ 
43.  Uniform  Summahility . — Following  analogy  with  uniformly  convergent 

series,  Hardy-^  has  proposed  the  following  definition  of  uniform  summability 

for  divergent  series,  basing  the  same  on  the  form  (IV)  (§  36)  of  definition  of  sum: 
Definition  I.     If  (instead  of  the  series  of  constant  terms  (11))  we  have  the 

series  (convergent  or  divergent) 

in  which  each  term  Unia)  is  a  function  of  the  (real)  variable  a,  this  series  is 
uniformly  summable  throughout  the  interval  jS  <  a  <  7  if  for  these  values  of  a 
the  integral 

00  /»00 

^Un{a)  =   I     e~''u{x,  a)dx 
0  Jo 


converges  uniformly,  wherein 


t(x,  a)  =  J2un(.ci) 


ni. 


I* 


Upon  the  basis  of  this  definition  the  following  theorems  analogous  to  those 
encountered  in  the  study  of  uniformly  convergent  series  may  be  established:^^ 
Theorem  I.     "  If  all  the  terms  Un{a)  are  continuous  functions  of  a  and 

IXl 

Sw„(q;) 

n=0 

is  uniformly  summable,  and 

Y.Un{(x)—. 

n=o  ni 

uniformly  convergent  for  any  finite  value  of  x,  in  an  interval  ((3,  7),  the  sum  of  the 
first  series  is  a  continuous  function  of  a  throughout  the  interval." 
Theorem  II.    "If 

00 

is  uniformly  summable  in  {ao  —  ^,  cxq-\-  ^)  a7id 

26  Cf.  Bromwich,  I.  c,  pp.  282-283. 

"  See  Transactions  Cambridge  Philos.  Soc,  Vol.  19  (1904),  p.  301. 

28  Cf .  Hardy,  I.  c. 


Uniform  Summability  97 


n=0  "•  • 


uniformly  convergent  for  any  finite  value  of  x,  the  series 

n=0 

may  be  differentiated  term  hy  term  for  a  =  ao" 
Theorem  III.     "If 

00 

(53)  Sunioc) 

71  =  0 

is  uniformly  summable  in  (/3,  7)  and 

I]  Wn(a)— j 
71=0  n  i 

uniformly  convergent  throughout  the  domain  (0,  X,  /3,  7)  however  great  he  X,  the 
series  may  he  integrated  term  hy  term  over  (/3,  7)." 

Extensions  of  Theorem  III  to  cases  in  which  (53)  fails  to  be  uniformly  sum- 
mable in  the  neighborhood  of  a  finite  number  of  isolated  points  within  (/3,  7) 
and  to  the  case  in  which  /3  =  00  have  also  been  obtained.  It  would  appear, 
however,  that  with  the  indicated  meaning  for 

00 
Sw7i(a), 

7!=0 

Theorems  I,  II  and  III  together  with  their  generalizations  relate  in  substance  to 
the  properties  of  definite  integrals  of  a  certain  prescribed  type  rather  than  to  the 
subject  of  infinite  series,  the  latter  appearing  merely  in  the  role  of  suggesting  the 
type  in  question.  For  this  reason  the  notion  of  "  uniform  summability,"  at 
least  as  formulated  upon  the  basis  of  definition  (IV)  (§  36),  together  with  the 
resulting  theorems  appear  somewhat  artificial.  This  seems  less  true,  however, 
in  case  definition  (I)  (or  (II))  is  adopted.  Thus,  confining  ourselves  for  sim- 
plicity to  the  important  case  in  which  r  =  1,  we  then  have  the  following 
Definition  IIP    A  series  (convergent  or  divergent) 

(54)  JlnM) 

n=0 

in  which  each  term  Un{oc)  is  a  function  of  the  (real)  variable  a,  is  uniformly 
summable  throughout  the  interval  /3  <  a  <  7  if  for  these  values  of  a  the  ex- 
pression 

5o(o;)  +  Si{a)  +  •  •  •  +  Sn{a) 


71+   1 


lohere         Sn(oc)  =  Voia)  +  Ui{a)  +  •  •  •  +  Un((x) 


converges  uniformly  to  a  limit  U(a). 

"  Cf.,  for  example,  C.  N.  Mooue,  Transaclioiis  American  Math.  Soc,  Vol.  10  (1909),  p.  400. 


98  Studies  on  Summability 

The  theorems  corresponding  to  I,  II  and  III  now  become  considerably  more 
direct.     Thus,  corresponding  to  Theorem  I  we  evidently  have  the  following: 

"  If  all  the  terms  ?/„(«)  of  the  series  (54)  are  continuous  and  the  same  series 
is  uniformly  summable  throughout  the  interval  ((3,  y),  then  its  sum  U{a)  is  con- 
tinuous throughout  (j3,  7)." 

The  corresponding  forms  for  Theorems  II  and  III  can  be  at  once  supplied. 

Supplementary  Remarks  and  Theorems 

44.  From  §§  41-43  it  may  be  concluded  that  of  the  six  definitions  of  "  sum  " 
in  §  36  those  deserving  of  especial  emphasis  are  (I)  (Cesaro)  or  its  equivalent 
(II)  (Holder)  and  (IV)  (Borel).  We  now  add  certain  noteworthy  results 
respecting  (I)  and  (IV),  omitting  proofs  in  cases  where  suitable  references  can 
be  given. 

1.  If  a  series  (convergent  or  divergent)  is  summable  by  Cesaro' s  method  for  a 
given  value  of  r  (cf.  §  36),  it  is  summable  by  the  same  method  for  all  larger  (inte- 
gral) values  of  r. 

In  fact,  with  *S„^''^  and  D^^''^  defined  as  in  (12),  we  have  the  identities 

^.C+l)   =    So('->  +   ^i^'-)  +   So^^^  +    •  •  •    +   Sn^^\ 
D,^r+1)    =    J)^ir)  _^   J)^ir)  _^   J)^^ir)  +    .  .  .    +   J)n^^\ 

and  since  by  hypothesis  lim  Sn^^'^Dn'-''^  exists,  it  follows  from  a  well-known 
theorem  due  to  Stolz''"  that  lim  Sn^'^^^/Dn^'^^^  also  exists  and  has  the  same 
value,  provided  however  that  as  n  increases  »Sn^''^  eventually  does  not  oscillate 
but  is  such  that  lim  »S„^'"^  =  ±  co  — a  condition  here  fulfilled  because  by  hypoth- 

n=oo 

esis  lim  <S„^''V-On^''^  exists,  while  from  the  definition  of  Z)„^''^  we  have  at  once 
limZ)„(^)  =  +  °o. 

2.  A  necessary  condition  that  any  series 

n=0 

be  summable  by  Cesaro' s  method  with  a  degree  of  indeterminacy  r  is  that 

(55)  lim  {un/n')  =  0.        ^i 

A  noteworthy  corollary  of  this  result  is  as  follows: 

3.  Let 

(56)  2  anX"" 

n=0 

3"  See  Math.  Annalen,  Vol.  33  (1889),  pp.  236-245. 
^'  For  a  proof,  see  Bromwich,  I.  c,  §  127. 


SUPPLEMENTAKY   REMARKS  99 

be  any  power  series  having  a  radius  of  convergence  equal  to  1.     Then  the  divergent 
series 

06 

n=0 

wherein  Xo  represents  any  special  value  such  that   |.ro|  >  1  cannot  be  summed  by 
Cesaro's  method.     Thus,  in  particular  Cesaro's  formula  cannot  serve  to  prolong 
analytically  the  power  series  (56)  outside  its  circle  of  convergence. 
In  fact,  placing  w„  =  a„a;o"  we  have 


and  hence 
Whence 


T  Ufl 

lim =  .To 

n=oo  ^n — 1 


— —  =  Xo-\-  €„;        hm  €n  =  0. 


Un  =  WoGto  +  eOCa-o  +62)  •  •  •  (a:o  +  €„). 


Now,  having  chosen  an  arbitrarily  small  positive  quantity  77,  we  have  |  Cn  |  <  ^ 
for  all  n  >  a  determinate  value  n^,  and  hence 

I  .^0  +  €n  I  >  I  Xo  I  —  I  r?  I  ;         n  >  nr,. 

Thus,  as  n  increases  indefinitely  the  expression  w„  becomes  infinite  to  as  high 
an  order  as  that  of  (|a-o|  —  It?!)"*.  But  for  a  sufficiently  small  choice  of  77  we 
have  |a:o|  —  [r/l  >  1,  since  by  hypothesis  |a-o|  >  1.  Thus  (55)  cannot  be  satis- 
fied for  any  value  of  r. 

In  contrast  to  this  result,  we  have  the  following  important  theorem  arising 
when,  instead  of  the  definition  (I)  of  sum,  we  adopt  the  definition  (IV)  of  Borel. 

4.  Let 

fix)    =   12  Ctn-T" 

be  any  power  series  having  a  radius  of  convergence  equal  to  1.     //,  then,  the  series 

71=0 

is  summable  by  definition  (IV)  (§  36)  so  also  is  the  series 

00 

S  aniTo" 

n=0 

provided  xq  lie  within  the  polygon  formed  by  tangents  to  the  given  circle  at  the  points 
{assumed  finite  in  number)  upon  the  circumference  at  ichich  f{x)  has  singularities. 
Moreover,  f{x)  may  be  extended  analytically  to  all  such  points  xq  by  mean^  of  the 
sum  formula  in  question,  i.  e., 


100  Studies  on  Summability 


/(•^o)  =    I     e~'^u{aXQ)da 

Jo 


where 

,      ,       ^aniax^Y 
u{oiX^  =  2^ ^ —  . 

n=0         ^  • 

r/ie  summahility  at  xq  will  be  absolute  (§  42)  anc^  i^  will  be  uniform  (§  43) 
throughout  any  region  situated  ivholly  within  the  indicated  polygon  (polygon  of 
summability).^^ 

5.  Absolutely  convergent  series  are  absolutely  summable,  but  series  that  are 
merely  convergent  may  not  be  absolutely  summableP 

6.  //  but  one  of  tico  series  is  absolutely  summable  while  both  are  summahle  by 
definition  (IV)  {BoreVs  integral)  to  the  respective  limits  Si,  S2,  then  the  product 
series  (cf.  (37))  is  summable  by  the  same  definition  to  the  value  Si,  S2,  but  not  neces- 
sarily absolutely  summable. ^^ 

7.  If  two  series  are  summahle  by  definition  (IV)  {BoreVs  integral)  to  the  values 
S\,  S2  respectively,  then  the  product  series  (cf.  (37))  whenever  summable  necessarily 
has  the  sum  sis^.^^ 

8.  If  the  coeficients  ui,  U2,  u^,  •••  of  the  divergent  series  (11)  are  such  that 
the  expressions 

Eo  =  Wo,         El  =  Uo-\-  ui, 

E2  =   Uq  +  2Ui  +  U2, 

/r  js  Es  =  Wo  +  3wi  +  3w2  +  W4, 


w(n  -1)     ,   ,  ,  , 

En  =  wq  +  nui  H ^-j —  u^-f-  •  •  •  +  nw„_i  +  w„ 

all  vanish  after  a  certain  point :  n  =  m,  then  the  series  may  be  summed  by  definition 
(IV)  {BoreVs  integral)  and  the  sum  will  be 

—  ^  A-  ~ -i-  ^  -\-  I       -^"t 

*~   2    '2-        2^  2"'''"^ 

— i.  e.,  the  sum  will  be  given  by  summing  the  series  by  Euler's  well-known 
method  for  converting  a  slowly  convergent  series  into  a  more  rapidly  converging 
one.''® 

'2  Proof  of  the  various  statements  here  made  is  readily  supplied  from  the  remarks  of  Brom- 
WICH,  I.  c,  §  113. 

33  Cf.  Hardy,  Quarterly  Journ.  of  Math.,  Vol.  35  (1904),  pp.  25,  28. 
3<  Cf.  Hardy,  I.  c,  pp.  43-44. 
3*  Cf.  Hardy,  I.  c,  pp.  44-45 
"  Cf.  Bromwich,  I.e.,  §  24. 


Supplementary  Remaeks  101 

This  result  evidently  becomes  of  especial  significance  for  all  series  (11)  of 
the  form 

«o  —  «!  +  052  —  «3  +  •  •  • ;         flm  positive 

for  which  the  successive  differences  between  the  quantities  Qq,  Oi,  oz,   •  •  •   all 
eventually  vanish  — e.  g.,  the  series 

1-2+3-4+5-  .-., 

wherein  the  quantities  ^o,  Ei,  E^,  etc.,  become 

J^o  =  1,  ^1  =  -  1,  E2  =  Es=   ■•■  =  En^O, 

and  hence  *  =  2  ~  i  =  i- 

The  proof  of  statement  (8)  may  be  readily  supplied  when  we  make  use  of 
the  Lemma  of  §  37.  Thus,  in  the  notation  there  employed,  let  us  take  in  the 
present  instance 

Sn       2  ^  22  ^  23  ^         ^  2"+i '        ""      ~  ~^  • 
Then 

7  T  Eq         El  ,      Em 

and  condition  (A)  of  the  lemma  is  at  once  seen  to  be  satisfied  (cf.  (16)). 
Application  of  the  lemma  thus  gives 

I  =  lim  e-^P  Z  '^H^  Sn  =  lim  e-'"5(2a)  =  lim  e-"5(a), 

p=co  n=0      ni  <i=<30  ^^^ 

where  s(a)  is  defined  by  (13).     Moreover,  this  result  may  be  written  (cf.  (25)) 
in  the  form 


/»00 

I  =  Uo-\-    I     e~''u'{a)da, 


where  u{a)  is  defined  by  (14).     If  the  integral  here  appearing  be  integrated  by 
parts  (cf,  (41))  we  thus  obtain 

/•oo 

I  =  -  [e-"M(a)]„=oo  +    I     e-''u{a)da. 

In  order  to  finish  the  proof  it  remains  but  to  show  that  the  first  term  here 
appearing  in  the  second  member  is  equal  to  zero. 

Upon  noting  the  meanings  of  Eq,  Ei,  E^,  •  •  •,  as  given  in  (57),  we  obtain 

euisx)  =e"(wo+WiQ;+W2^+  •••)  =  J^o  +  i?i«  +  ii^a  m  +  •••  +  ^m— , 
and  hence 

lim  e-M(a;)  =  lim  e-2«  XE^^E^a^ h  i^m  — ,  1  =  0. 

a=oo  a=»  L  ml  J 


CHAPTER  V 

THE  SUMMABILITY  AND   CONVERGENCE   OF  FOURIER  SERIES  AND   ALLIED 

DEVELOPMENTS 

45.  In  the  present  chapter  it  is  proposed  to  derive  the  principal  known 
results  concerning  the  summability  of  Fourier  series  and  other  allied  develop- 
ments for  functions  of  one  real  variable  (developments  in  terms  of  Bessel  func- 
tions, Legendre  functions,  etc.).^  We  shall  take  the  word  "  sum  "  in  the  Holder 
sense-  (§  36)  according  to  which  a  given  series  (convergent  or  divergent) 

(1)  flun 

n=0 

has  its  sum  s  defined  by  the  equation 

(2)  s  =  lim  5„^''^;        r  =  fixed  integer  ^  0, 
where 


(0) 


=   5„  =   Wo  +  Wl  +    •  •  •   +  U„ 


n  +  1 

Moreover,  if  the  terms  w„  are  functions  of  the  (real)  variable  x  (as  will  now 
always  be  the  case)  when  considered  throughout  an  interval  (a,  h),  the  series  (1) 
will  be  termed  uniformly  summable  throughout  (a,  h)  in  accordance  with  the  defi- 
nition n  of  §  43  — i.  e.,  provided  that  the  limit  (2)  is  approached  uniformly  for 
the  same  values  of  x. 

In  view  of  the  fact  that  any  discussion  of  the  summability  of  Fourier  series 
and  other  allied  developments  is  intimately  connected  with  the  corresponding 
discussion  of  convergence,  the  latter  being  in  fact  but  the  case  of  summability 
in  which  r  =  0  (cf.  (2)),  we  shall  as  a  matter  of  course  elaborate  both  aspects 
of  the  subject.^  No  attempts  will  be  made  however  to  obtain  theorems  con- 
taining the  minimum  restrictions  for  a  given  function  j{x)  in  order  that  it  be 

^  See  explanatory  remarks  in  the  Preface. 

"^  The  results  obtained  will  therefore  (§  38)  be  convertible  at  any  point  into  those  for  summa- 
bility in  the  CesA.ro  sense. 

^  Since  all  convergent  series  are  summable  but  not  conversely  it  is  evident  that  more  restric- 
tive conditions  upon  w„  are  in  general  necessary  to  insure  convergence  than  summability.  This 
fact  will  be  well  illustrated  in  the  studies  of  the  present  chapter. 

102 


Theoeems  on  Fourier  Series  103 

developable  in  a  summable  (or  convergent)  series  of  any  one  type.  The  emphasis 
will  be  placed  rather  upon  the  attainment  of  a  general  theory  of  such  a  nature 
that  the  various  more  important  special  developments,  including  Fourier  series, 
and  the  familiar  developments  in  terms  of  Bessel  functions  and  Legendre 
functions,  may  be  studied  as  special  applications  of  it,  provided  f{x)  satisfy  any 
one  of  various  slightly  limiting  conditions."^  This  general  theory  is  elaborated 
in  §§  46-56  following  which  the  applications  just  mentioned  have  been  carried 
through  (§§  57-70). 

The  basis  of  the  entire  chapter  is  Dini's  great  work  entitled  "  Serie  di  Fourier 
e  altre  rappresentazioni  analitiche  delle  funzioni  di  una  variabile  reale  "  (Pisa, 
1880)  and  due  acknowledgment  is  here  made  to  this  source. 

I 

Fourier  Series 

46.  If  f{x)  be  a  given  function  of  the  real  variable  x  defined  throughout  the 
interval  (—  tt,  tt)  the  corresponding  Fourier  series  is  by  definition 

00 

(3)  |ao  +  zZ  (cfn  cos  nx  +  6„  sin  nx), 

M  =  l 

where 

an=       \    fix)  cos  nxdx,        K  =  -  \    fix)  sin  nxdx. 

As  regards  the  convergence  and  summability  of  this  series,  the  following 
theorems  are  well  known: 

Theorem  I.  If  f(x)  remains  finite  throughout  the  interval  (—  tt,  tt)  with  the 
'possible  exception  of  a  finite  number  of  points  and  is  such  that  the  integral 


(4)  r  \f{x)\dx 

%)  —  TT 


exists,  then  the  Fourier  series  (3)  will  converge  at  any  point  x  (—  ir  <  x  <  t)  in 
the  arbitrarily  small  neighborhood  of  ichich  f{x)  has  limited  total  fluctuation,  and 
the  sum  will  be 

H/(^-o)+/(.T  +  o)]. 

Moreover,  the  convergence  will  be  uniform  to  the  limit  f{x)  throughout  any  in- 
terval (a',  b')  inclosed  within  a  second  interval  (oi,  bi)  such  that 

-  IT  <  ai  <  a'  <  b'  <  bi  <  IT 

<As  regards  convergence,  including  uniform  convergence,  general  theorems  of  the  nature 
here  indicated  together  with  applications  have  been  given  by  Hobson  in  a  series  of  memoirs 
appearing  in  the  Transactions  of  the  London  Math.  Society  (Vol.  6  (1908),  pp.  349-395;  Vol.  7, 
pp.  24-48;  ibid.,  pp.  338-388).  Corresponding  general  studies  for  summability  do  not  appear 
to  have  been  thus  far  carried  through,  though  numerous  results  have  been  obtained  by  special 
methods.     For  further  remarks,  see  notes  appended  to  the  theorems  of  §§67  and  68. 


104  SUMMABILITY   OF   FOURIER   SeEIES   AND   ALLIED    DEVELOPMENTS 

provided  thatf{x)  is  continuous  throughout  (a',  h')  inclusive  of  the  end  points  x  =  a', 
X  =  b'  and  has  limited  total  fluctuation  throughout  (ai,  61).^ 

Theorem  II.  If  f{x)  remains  finite  throughout  the  interval  (—  tt,  tt)  with  the 
possible  exception  of  a  finite  number  of  points  and  is  such  that  the  integral  (4)  exists, 
then  the  Fourier  series  (3)  will  be  summable  (r  =  1)  at  any  point  x  {—  ir  <  x  <  ir) 
at  which  the  limits  f{x  —  0),  f{x  +  0)  exist,  and  the  sum  will  be 

i[/(^_0)+/(.r+0)]. 

Moreover,  the  summability  ivill  be  uniform  to  the  limit  f(x)  throughout  any 
interval  (a',  b')  such  that  —  t  <  a'  <  b'  <  ir  provided  that  f(x)  is  continuous 
throughout  {a',  b')  inclusive  of  the  end  points.^ 

Theorem  III.  If  f{x)  when  considered  throughout  the  interval  (—  tt,  tt)  satis- 
fies the  conditions  mentioned  in  Theorem  I  and  is  such  that  in  arbitrarily  small 
neighborhoods  at  the  right  of  the  point  x  =  —  tt  and  at  the  left  of  the  point  x  =  tt 
it  has  limited  total  fiuctuation,  then  the  Fourier  series  (3)  will  converge  when  x  =  —  w 
or  X  =  TT  and  in  either  case  the  sum  will  be 

H/(^-o)+/(-x  +  o)]. 

Theorem  IV.  If  f{x)  when  considered  throughout  the  interval  (—  tt,  tt)  satis- 
fies the  conditions  mentioned  in  Theorem  I  and  is  such  that  the  limits  /(tt  —  0), 
/(—  TT  +  0)  exist,  then  the  Fourier  series  (3)  will  be  summable  lohen  x  =  ir  or 
X  =  —  TT  and  in  either  case  the  sum  will  be 

H/(7r-0)+/(-7r+0)]. 

It  is  our  purpose  here  (having  in  mind  the  essential  steps  incident  to  the 
formation  of  a  general  theory  for  the  study  of  this  and  other  allied  develop- 
ments) to  show  in  the  first  place  that  the  proof  of  Theorem  I  may  be  made  to 
depend  upon  the  existence  of  the  three  following  relations  which  themselves  are 
independent  of  the  function  f{x)  and  concern  only  the  trigonometric  expression 

.    2n±l^ 
sm  — ^ —  t 

(5)  (p{n,  t)  =  ~  , 

27r  sin  ^ 

n  being  limited  to  positive  integral  values. 
(I)  The  integral 

(fill,  t)dt 


^Cf.  HoBSON,  "Theory  of  Functions,"  §§448,  451,  457,  459.     Also,  Chapman,  Quarterly 
Journ.  of  Math.,  Vol.  43  (1911),  p.  33. 
6  Cf.  HoBSON,  I.  c,  §  469. 


Proof  of  Theorems  105 

when  considered  for  values  of  t  in  the  interval  —  27r +€<<<  —  e,  e  being  an 
arbitrarily  small  positive  constant,  converges  uniformly  to  the  limit  —  |  when 
n  =  CO  ;  while  the  same  integral  when  considered  for  values  of  t  in  the  interval 
e  =  i  =  27r  —  e  converges  uniformly  to  the  limit  ^  when  n  =  co  J 

(II)  For  a  sufficiently  small  choice  of  the  positive  quantity  e  we  have 


'Jo 


(p(n,  t)dt 


<  A;         -  e^t^e, 


where  ^  is  a  constant  independent  of  both  n  and  t.  ^ 

(III)  For  a  sufficiently  small  choice  of  the  positive  quantity  e  we  have 

I (p(n,  t)\<  B;         -  2Tr  +  €  ^  t  ^  -  €,         e  ^  t  ^  2Tr  -  e, 

where  5  is  a  constant  independent  of  both  n  and  t. 

In  order  to  prove  that  Theorem  I  depends,  as  stated  above,  upon  the  existence 
of  these  three  relations,  let  us  suppose  at  first  that  x  has  some  special  value  x  =  a 
such  that  —  X  <  a'  ^  a;  ^  6'  <  TT,  the  quantities  a',  h'  being  regarded  as  fixed. 
With  this  value  of  a  the  {n  +  l)st  term  of  the  series  (3)  takes  the  form 

~  I     /('^)(cos  Tix  cos  na  +  sin  nx  sin  na)dx  =  -   I     f{x)  cos  n{x  —  a)dx, 

SO  that  the  sum  of  the  first  (n  +  1)  terms  becomes 

Sn{a)  =  -   I     fix)  H  +  Zl  cos  n{x  —  a)  \  dx. 


Upon  making  use  of  the  well-known  relation 

n 

+  zl  cos  nx  = 


.    2n  +  1 
sm  — - —  X 


1 


71=1  n      •      "^ 

2  sinr 
we  thus  have 
(6)  Sn((x)  =    I     f{x)(p{n,  X  -  oi)dx, 

J  —-n 

where  (p{n,  x  —  a)  is  to  be  determined  by  (5). 

Whence  also,  having  chosen  an  arbitrarily  small  positive  quantity  e,  we  may 
write 


(7) 


f{x)(p{n,  X  —  a)dx  +    I     j{x)<p{n,  x  —  a)dx 

+    I     f{x)(p{7i,  X  -  a)dx  +    I        fix) (fin,  x  -  a)dx. 

Ja—e  *Ja 


''  For  a  proof  of  this  statement,  see  Appendix,  §  1 . 
^  For  a  proof,  see  Appendix,  §  2. 


106  SOIMABILITY   OF   FOUKIER   SERIES   AND   AlLIED    DEVELOPMENTS 

We  may  now  show  that  the  conditions  placed  upon  f{x)  for  the  whole  (closed) 
interval  (—  tt,  tt)  (cf.  Theorem  I)  when  taken  in  conjunction  with  relations  (I) 
and  (III)  suffice  to  make  the  limit  approached  by  each  of  the  first  two  integrals 
of  (7)  equal  to  zero  when  n  =  oo ,  In  fact,  we  shall  show  that  this  limit  is 
approached  uniformly  by  each  of  these  two  integrals  when  they  are  considered 
for  values  of  a  for  which  a'  ^  a  ^  b'. 

Considering,  then,  the  first  integral  in  the  second  member  of  (7),  let  us  repre- 
sent by  Xi,  xi,  Xz,  '  •  •,  Xq  (Xs  >  a^s-i)  the  points  (q  in  number)  at  which  f{x) 
becomes  infinite  in  the  (closed)  interval  (—  tt,  tt),  assuming  at  first  for  simplicity 
that  Xi  ^  —  TT,  Xq  ^  TT.  Having  chosen  an  arbitrarily  small  positive  quantity 
CO,  let  us  also  suppose  at  first  that  the  value  x  =  a  —  e  lies  within  one  of  the 
following  intervals: 

(8)  (—  IT,  Xi—  0)),         (.Ti  +  CO,  a'2  —  co),  •••,         (a:g+co,  —  x), 

i.  e.,  let  us  assume  that  a:  =  a  —  e  is  not  one  of  the  points  at  which  f{x)  becomes 
infinite.     We  may  then  express  the  integral  in  question  in  the  form 

(9)  1       f(x)cp(n,  X  -  a)dx  ^  S  +  R, 
where 

+  +  •  •  •  +  +  ]f{x)^{n,  X  -  a)dx,         (g  ^  q) 

and 

+  +  •  •  •  +  )  f{x)<p{n,  X  -  a)dx. 

Now,  introducing  relation  (III),  we  have 

'         +  +  •••+  ]\m\dx 

and  since  the  integral  (4)  is  assumed  to  exist  it  thus  follows  that  for  a  sufficiently 
small  choice  of  co  we  shall  have  \R\<  Bgp  <  Bqp  where  /o  is  a  preassigned 
arbitrarily  small  positive  constant. 

The  value  of  p  having  been  assigned  and  co  then  determined  in  the  manner 
just  indicated,  we  turn  to  the  expression  S.  In  considering  this  it  is  first  desirable 
to  make  the  following  observation. 

Consider  the  set  of  intervals  (8) .  Let  us  divide  the  first  of  these  into  y  equal 
sub-intervals  of  length  6i,  the  second  into  the  same  number  j)  of  equal  sub- 
intervals  52,  •  •  •,  the  {q  +  l)st  into  the  same  number  y  of  equal  sub-intervals 
of  length  5^+1.  Let  Di,  g  be  the  fluctuation  oi  f{x)  in  the  sih  one  of  the  intervals 
5i,  let  Z>2,  s  be  the  fluctuation  of  f{x)  in  the  5th  one  of  the  intervals  62,  •  •  •,  let 
Dq+i,s  be  the  fluctuation  of /(.r)  in  the  5th  one  of  the  intervals  dq+i.  Finally, 
let  us  form  the  sums 


Proof  of  Theorems  io7 


(10)  hilDr,,,        hJlD,,,,        ...,        5,+ii:Z) 

«— 1  s=l  .-1 


«=1 


3+1,  8- 


Since /(.t)  is  integrable  over  each  of  the  intervals  (8),  it  follows  that  we  can  make 
our  choice  of  the  integer  p  so  large  that  each  of  the  sums  (10)  will  be  less  in 
absolute  value  than  the  preassigned  quantity  p  already  mentioned.  At  the 
same  time,  y  may  be  chosen  so  large  that  each  of  the  integrals 

(11)  (  \f{x)\dx;        ,=  1,2,3,  ...,9+1, 

where  the  integration  is  performed  over  any  one  of  the  intervals  5„  will  like\^^se 
be  less  than  p.  In  what  follows  the  quantity  y  will  be  understood  to  be  any 
special  one  determined  according  to  the  two  conditions  just  indicated. 

Returning  to  the  expression  S,  let  us  now  consider  the  first  of  the  integrals 
of  which  it  is  constituted.  Calhng  x  =  ^,_i,  x  =  ^,  the  values  of  x  corresponding 
to  the  end  points  of  the  5th  one  of  the  intervals  5i,  we  have 

'J-n  s=iJ^,_i  [  (p  =  (p{n,  x  —  a). 

Now,  introducing  the  constant  B  defined  in  (III),  we  may  write 

f'f'<pdx=    f'f-(<p+  B)dx  -  B  C'fdx 

and  the  function  (p  +  B  will  be  positive  for  all  values  of  x  such  that  ^^-i  <  x  <  ^^ 
(n  having  any  value  which  it  may  take).  Hence,  upon  applying  the  first  law  of 
the  mean  for  integrals,  we  have 

r  '/  •  <pdx  =  f,  f\dx  +  Bf,  f'dx-  Bf/  C'dx, 
•^^-1  '^f.-i  ^f.-i  »/f.-i 

where  fs  and  //  are  certain  values  lying  between  the  upper  and  lower  limits  of 
f(x)  when  ^s-i  <  x  <  ^s- 

Since  ^s  —  ^s-i  =  5i  we  thus  have 

I     f  '  (pdx  =  fs  I     (pdx  +  dsBdiDi,  s, 

where  ds  is  a  quantity  lying  between  —  1  and  +  1  and  where  Z)i, ,  has  the  meaning 
already  indicated. 

Hence,  recaUing  what  has  been  said  of  the  sums  (10),  we  may  write 

(12)  r       f  ■  <pdx=j^fA'  ipdx+eBp;         -1<^<1. 


108  SUMMABILITY   OF   FOURIER   SeRIES   AND   AlLIED   DEVELOPMENTS 

Now, 

I     (pdx  =    I         (p(jn,  t)dt  =    I         <;i?(n,  <)<i<  —    I  (p{n,  t)dt 

and  corresponding  to  a  second  arbitrarily  small  positive  quantity  (x  we  may, 
by  virtue  of  relation  (I),  find  a  quantity  n„  independent  of  a.  such  that 


f 

'Jo 

f 

Jo 


^{n,  t)dt  =  -  i  +  eicr 


(p{n,  t)dt  =  —  2  +  620* 


Whence, 


^c?a; 


<    2(7 


n  >  n„, 

-  1<  01  <  1, 

a'  <  a  <  h'. 


n  >  n„, 

a'  <a<  h', 


1<  62  <  1, 


and  hence  also  (cf.  (12))  we  have  for  the  value  of  a  under  consideration 

»fl— <o 


(13) 


L  ^ 


(pdx   <  2Mpa  -\-  Bp]        n  >  n^, 


where  M  represents  the  upper  limit  of  |/(.r)  |  in  the  intervals  (8). 

Similarly,  all  the  g  -\-  \  constituent  integrals  of  S,  except  the  last,  may  be  thus 
treated,  thereby  leading  to  the  equation 


(14) 


S  =  P  + 


/^a—e 

I        /  •  (pdx, 


where  for  all  values  of  n  greater  than  some  value  independent  of  a  we  have 

I P I  <  2gMp(x  +  gBp  ^  2qMpa-  +  qBp. 

Let  us  consider  finally  the  integral  appearing  in  (14).  For  this  we  first  note 
that  the  interval  of  integration  consists  of  a  portion  (or  at  most  the  whole)  of 
the  interval  {xg  +  co,  Xg+i  —  co)  belonging  to  the  set  (8).  Let  us  suppose  that 
rjr  <  oi  —  €  ^  r)r+i  where  rjr  and  r]r+i  are  the  values  of  x  corresponding  to  the 
extremities  of  the  rth  of  the  p  divisions  of  length  8g+i  into  which  we  have  already 
divided  the  interval  {xg  +  co,  Xg+i  —  co).     We  may  then  write 

I       /  •  (pdx  =1       /  •  (pdx  +1       /  •  (pdx. 

The  last  integral  here  appearing  is  less  in  absolute  value  (cf.  relations  (III) 
and  (II))  than 

(15)  B  r  '  \f(x)\dx  <  B  P"  \f(x)\dx  <  Bp 

where  p  has  the  meaning  already  given. 


Proof  of  Theorems  109 

Again,  let  there  he  I  {I  "^  p)  of  the  divisions  8g  in  the  interval  {Xg  +  co,  r]r). 
Then,  treating  the  first  integral  in  the  second  member  of  (15)  as  we  did  the  first 
integral  in  S,  we  obtain  (cf.  (13)) 


I       /  •  (pdx 


<  2Mh  +  Bp^  2Mp(7  +  Bp;        n  >  n„, 


where  n^^  is  independent  of  a. 

In  summary,  then,  we  have  the  following  result:  Let  xi,  xi,  x^,  -  •  ■ ,  Xs,  •  ■  - ,  Xq', 
(Xs  >  Xs-i);  {xi  4=  —  TT,  a:,  =#  x)  represent  the  q  points  within  the  interval 
(—  TT,  tt)  at  which /(x)  becomes  infinite,  and  let  a  be  any  value  such  that  a  —  e 
(cf.  (7))  lies  within  one  of  the  intervals 

(—  TT,  Xi   —   Oi),  (.Ti  +   CO,  .T2   —   Co),  •■•,  (Xg  +   CO,  Tt)  ; 

03  arbitrarily  small  and  positive 

and  also  such  that  —  x  <  a'  ^  a  ^  b'  <  tt.  Then,  corresponding  to  an  arbi- 
trarily small  positive  quantity  p  and  a  second  such  quantity  a,  we  may  determine 
a  positive  value  n^  independent  of  a  and  such  that 


£ 


f(x)cp(n,  X  -  a)dx    <  2pM(q  +  1)(t  +  B{q  +  l)p;         n  >  n. 


Since  B,  q,  M  and  j)  as  well  as  n^  are  each  independent  of  a,  it  follows  that  for 
all  the  indicated  values  of  a  the  first  integral  in  the  second  member  of  (7)  converges 
uniformly  to  zero  when  n  =  oo . 

It  remains  to  show  that  the  same  is  true  when  a  —  e  pertains  to  one  of  the 
intervals  of  the  following  set : 

{xi  —  CO,  Xi-\-  co),         {X2  —  0},  X2  +  co),         '  ",         (x g  —  CO,  X q -\-  oi) ; 

03  arb.  small  and  positive. 

The  desired  result  follows  by  reasoning  directly  analogous  to  the  preceding 
after  rewriting  (9)  in  which  S  and  R  are,  however,  defined  as  follows: 

+  +  •  •  •  +  f{x)<p{n,  X  -  a)dx,         (g  ^  q), 

+  +  •••+  +  )f{x)<p{n,x-a)dx. 

Again,  the  same  conclusion  may  be  likewise  reached  in  case  either  or  both 
of  the  points  x  =  —  tt,  x  =  ir  are  points  at  which  /(.t)  becomes  infinite.  The 
forms  in  which  S  and  R  should  then  be  taken  readily  suggest  themselves  and  are 
therefore  suppressed. 


110  SUMMABILITY   OF   FoUEIER   SeRIES   AND   AlLIED    DEVELOPMENTS 

In  like  manner  it  appears  that  the  second  integral  in  the  second  member  of 
(7)  converges  uniformly  to  zero  when  w  =  co  for  all  values  of  a  such  that 

—  TV  <  a'  ^  a  ^h'  <  -K. 

These  results  having  been  established,  we  turn  to  a  consideration  of  the  last 
two  integrals  in  the  second  member  of  (7).  We  shall  suppose  at  first  that  a 
has  any  sjjecial  value  such  that  —  tt  <  a'  =  a;  =  6'  <  tt. 

Since  by  hypothesis  f{x)  is  of  limited  total  fluctuation  in  the  neighborhood  of 
the  point  x  =  a,  the  expressions  f(a.  —  0),  f{a  +  0)  certainly  have  a  meaning.^ 
We  may  therefore  write  the  third  integral  in  the  second  member  of  (7)  in  the  form 


(16) 


/(a  -0)J_  <pin,  t)dt  +  j_[f{a+t)-  f{a  -  0)]cp{n,  t)dt. 


Wlien  n  =  00  the  first  term  here  appearing  approaches  the  limit  \f{a  —  0) 
as  a  result  of  relation  (I).  As  to  the  second  term,  it  follows  from  our  hypotheses 
upon  f{x)  in  the  neighborhood  of  the  point  x  =  a  that  the  function  /(a  +  t) 
—  f(a  —  0)  is  of  limited  total  fluctuation  in  the  interval  —  e  <  t  <  0,  a,t  least 
if  e  be  chosen  sufficiently  small.  Whence,  in  this  interval  the  same  function 
will  be  either  monotone  or  will  consist  of  the  difference  of  two  monotone  func- 
tions.^*' In  the  former  case  we  may  apply  the  second  law  of  the  mean  for  integrals 
and  write 

(17)     J"  [/(a  +t)-  f{a  -  0)Mn,  t)dt  =  [f{a  -  e)  +  f{a  -  0)]  J  ''  ^{n,  t)dt; 

0  <  ei  <  e. 

At  the  same  time  our  choice  of  e  may  be  made  so  small  that  the  expression 
\f{a  —  e)  —  f{a  —  0)  |  will  be  less  than  any  preassigned  quantity  a.  With  e 
thus  chosen,  we  have  now  but  to  make  use  of  relation  (II)  to  see  that  the 
second  term  of  (16)  may  be  made  less  in  absolute  value  than  Aa,  whatever  the 
value  of  n.  In  case  f{a  -\-  t)  —  f{a  —  0)  consists  of  the  difference  of  two  mono- 
tone functions,  the  proof  may  evidently  be  carried  out  in  a  similar  manner, 
showing  that  in  this  case  the  absolute  value  in  question  will  be  less  than  2A(t. 

Therefore,  the  limit  of  the  sum  of  the  first  and  third  terms  in  the  second 
member  of  (7)  as  w  =  co  is  f/(a;  —  0).  Similarly,  the  limit  approached  by  the 
sum  of  the  second  and  last  terms  is  ^f{a  +  0). 

The  first  part  of  Theorem  I  is  thus  fully  established.  It  remains  to  consider 
only  that  part  which  concerns  uniform  convergence,  and  since  we  have  already 
shown  that  for  all  values  of  a  such  that  — Tr<  a' ^a^b'<T  the  first  and 
second  terms  in  the  second  member  of  (7)  converge  uniformly  to  zero,  it  will 

9  Cf.  HoBsoN,  I.  c,  §  194. 
'0  Cf.  HoBSON,  I.  c,  §  195. 


Proof  of  Theorems  HI 

now  suffice  to  show  that  under  the  hypotheses  of  the  last  part  of  the  Theorem 
the  last  two  terms  of  (7)  when  considered  for  the  same  values  of  a  each  converge 
uniformly  to  the  limit  ^/(a). 

Now,  if  f{x)  be  continuous  (as  the  present  hypotheses  demand)  throughout 
the  interval  (a',  h')  {x  =  a' ,  x  =  h'  included)  then/(.r)  will  be  uniformly  continu- 
ous throughout  this  interval.^^     Hence,  corresponding  to  an  arbitrary  choice  of 

11  Cf.  HoBSON,  I.  c,  §  175. 
the  positive  quantity  a,  it  is  possible  to  determine  a  positive  e  independent  of  a 
and  such  that 

(18)  |/(a  -  e)  -  /(«)  I  <  (t;         a'  <a<h'. 

Introducing  this  choice  of  e  into  (16),  we  may  again  write  (17)  for  all  the 
indicated  values  of  a  (a'  ^  a  ^  h')  since,  from  the  hypotheses  of  the  second 
part  of  the  Theorem,  it  follows  as  before  that  the  function  f{a  -\-  t)  —  f{a)  is 
either  monotone  or  consists  of  the  difference  of  two  monotone  functions  of  t 
throughout  the  interval  —  e  <  t  <  0  whatever  the  value  of  a  (a'  ^  a  ^  h') — 
at  least  if  e  be  taken  so  small  that  a'  —  e  >  ai,  where  ai  has  the  meaning  given 
in  the  Theorem. 

Thus,  we  reach  the  desired  result  respecting  the  third  term  in  the  second 
member  of  (7)  and  similarly,  we  reach  the  indicated  result  for  its  last  term. 

47.  We  turn  to  the  proof  of  Theorem  II.  It  is  our  purpose  here  to  show  that 
relations  (I)  and  (III)  of  §  46  together  with  the  following  suffice  for  the  proof: 

(II)'  Having  placed 

(19)  Hn,t)  =-^.  12^{n,t), 

n  -\-  1  n=0 

where  (p(n,  t)  is  the  trigonometric  expression  (5),  we  may  write  for  a  given  value 
of  the  positive  quantity  e  and  all  subsequently  chosen  sufficiently  large  values 
of  n 


£ 


\^{n,  t)\dt<  C; 


where  C  is  a  constant  (independent  of  both  n  and  e). 

In  proving  Theorem  II  we  shall  therefore  substitute  relation  (II)'  for  relation 
(II)  of  §  46,  but  we  shall  employ  relations  (I)  and  (III)  as  before. 

Assuming  first  that  a  has  any  special  value  such  that  —  it  <  a'  ^  a  ^  b'  <  tt, 
we  have  from  (7) 

^,    ^  [■^o(a)  +  si{a)  +  . .  .  4-  Sn{a)]  =    I        f(.v)^{n,  x  —  a)dx 
(20) 

+  I     f{x)^{n,  x—a)dx-\-  I     f{x)^(n,  x  —  a)dx  +  1       f{x)^(n,  x  —  a)dx, 

where  $  is  given  by  (19). 


112  SUMMABILITY   OF   FoURIER   SERIES   AND   AlLIED    DEVELOPMENTS 

Now,  the  fact  that  <p  satisfies  (I)  and  (III)  enables  us  to  say  at  once  that  <l» 
also  satisfies  the  same  relations.  In  fact,  if  ip  satisfies  (I)  the  principle  of  con- 
sistency (§  37)  as  applied  to  the  Holder  method  of  summation  shows  that  $ 
also  satisfies  it,  while  (III)  becomes  satisfied  by  $  since  we  may  write 

\^{n,t)\^-^\\<p{3hf)\^\ip{ii-  1,  01  +  •••  +k(0,  0|]  <  V=  ^• 

The  second  member  of  (20)  is  the  same  as  that  of  (7)  except  for  the  substi- 
tution of  <J>  for  ip,  and  since  <E>  satisfies  relations  (I)  and  (III)  it  follows  precisely 
as  in  the  discussion  in  §  46  that  the  first  two  integrals  on  the  right  in  (20),  when 
considered  for  values  of  a  such  that  —  tt  <  a'  ^  a:  =  6'  <  tt,  converge  uni- 
formly to  zero  as  n  =  co,  provided  merely  that  the  integral  (4)  exists.  The 
third  term  of  (20)  may  be  written  in  the  form 

/(a  -  0)  j    ^{n,  t)dt  +  J  [/(a  +  0  -  /(«  -  0)Mn,  t)dt, 

provided  that  f(a  —  0)  exists.  When  n  =  <x>  the  first  term  here  appearing 
approaches  the  limit  |/(q;  —  0)  since,  as  already  pointed  out,  $(??,  t)  satisfies  (I). 
As  to  the  second  term,  we  may  choose  e  so  small  that  throughout  the  interval 
—  6  <  i  <  0  we  shall  have  \fia  -{-  t)  —  f{a  —  0)\<  <x  where  a  is  an  arbitrarily 
small  preassigned  positive  quantity.  With  e  thus  chosen  and  n  then  taken 
suflBciently  large  the  term  in  question  becomes  less  in  absolute  value  than 

(21)  a  j     \^{n,t)\dt<  Co, 

where  C  is  the  constant  defined  in  connection  with  relation  (II)'.  Thus,  the 
sum  of  the  first  and  third  terms  of  (20)  approaches  the  limit  \j{a  —  0)  when  w=  co . 
Likewise,  the  sum  of  the  second  and  last  terms  of  (20)  is  seen  to  approach  the 
limit  \j{a  +  0). 

The  first  part  of  Theorem  II  thus  becomes  established,  and  in  order  to  prove 
the  second  part  it  suflSces  to  note  (cf.  the  discussion  of  (16))  that  if  /(.r)  is  con- 
tinuous throughout  the  interval  {o! ,  b')  inclusive  of  the  end  points,  then  the 
quantity  e  in  (20)  may  be  chosen  independently  of  a  (a'  <  a  <  b'). 

48.  Having  shown  that  Theorem  I  results  from  relations  (I),  (II)  and  (III) 
and  that  Theorem  II  results  from  (I),  (II)'  and  (III),  we  shall  now  show  that 
Theorem  III  results  from  (I),  (II)  and  (III)  together  with  the  following: 

(IV)  <p{n,  t±2Tr)  =  <p{n,  t).  ^^ 

Let  us  take  first  the  case  in  which  x  =  tt.     The  expression  for  Sniir)  may  be 
•2  The  proof  of  (IV)  is  immediate  from  (5). 


Pkoof  of  Theorems  113 

obtained  by  placing  a  =  -w  in  (6).     This  expression,  after  placing  x  —  it  =  t, 
becomes 


(22) 


Sn(T)  =  j    fiiT  +  t)<p{n,  t)dt  =  (  /  ^     +  J    )/(^  +  t)^in,  t)dt. 


Of  the  two  integrals  here  appearing  in  the  last  member,  the  first,  after  making 
the  substitution  t'  =  2^  -\-  t  and  dropping  accents,  takes  the  following  form  as  a 
result  of  (IV) 

r  /(-  TT + i)<pin,  t)dt. 

Jo 
Whence,  we  may  write 

*n(7r)  =    r  V(vr  +  t)<p{n,  t)dt  +    C f{-  tt  +  t)<p{7i,  t)dt 
(23)  0  6  (t  >  0)• 

+   r  /(tt  +  0«p(^i,  0^^  +    r  /(-  vr  +  Ov(^,  0^^ 

We  may  now  show  that  as  n  =  oo  the  limit  approached  by  each  of  the  first 
two  integrals  here  appearing  is  0.  In  order  to  do  this  it  will  suffice,  since  the 
integral  (4)  exists,  to  show  that  the  property  just  indicated  is  true  of  each  of  the 
integrals 

J  /(tt  +  t)cp{n,  t)dt,         J  /(-  TT  +  t)<p{n,  t)dt, 

where  it  is  understood  that  /(tt  +  t)  remains  finite  throughout  the  (closed) 
interval  (c,  d);  — ir^c<d^  —  e,  while  f{—ir-\-t)  remains  finite  through- 
out the  (closed)  interval  (e,  f);  e  ^  e  <  /  ^  tt. 

Let  us  divide  the  interval  (c,  d)  into  p  equal  sub-intervals  each  of  length  5 
by  means  of  the  points  t  =  c,  t  =  t\,  t  =  U,  ■  -  - ,  t  =  tp-i,  t  =  d.  Then,  with 
the  meaning  for  B  appearing  in  (III),  we  may  write 

f'fia  +  tMn,  t)dt  =    i  V(«  +  t)[<p{n,  t)  +  B]dt  -  B   f  /(«  +  t)dt 

and  {(p{n,  t)  +  B]  will  be  positive  when  ts-\  <  t  <  ts  {n  having  any  fixed  value). 
Hence,  applying  the  first  law  of  the  mean  for  integrals,  we  obtain 

r  V(a  +  t)^(:n,  t)dt  =  fs  f'<pin,  t)dt  +  Bfs  f'dt  -  Bf/  f'dt, 

where  /«  and  fs  are  certain  quantities  lying  between  the  upper  and  lower  limits 
of  f{a  +  t)  when  ts-\  <  t  <  tg. 

Since  ta  —  ts-i  =  5,  we  thus  have 

f'fia  +  t)(p{7i,  t)dt  =  /,   r <p{n,  t)dt  +  d.BbD,;         -  1  <  0,  <  1, 

where  Z),  is  the  fluctuation  of /(a  +  t)  in  (/«_!,  tg). 
9 


114  SUMM ABILITY   OF   FoUKIER   SERIES  AND   ALLIED   DEVELOPMENTS 

Hence  also 

f{a  +  t)(p{n,  t)dt  =  T.fs        <p{n,  t)dt  +  055  Z^al     -  1  <  0  <  1. 

Now,  by  taking  p  sufficiently  large  the  last  term  of  (24)  may  be  made  arbi- 
trarily small  in  absolute  value,  as  follows  from  the  existence  of  (4).  The  value 
of  p  having  once  been  chosen,  let  us  allow  n  to  increase  indefinitely.  The  last 
term  of  (24)  continues  arbitrarily  small  in  absolute  value,  while  its  first  term 
approaches  the  limit  zero,  as  appears  directly  upon  writing 

*}t^x        Jo  Jo 

and  applying  (I). 

Similarly,  the  second  term  in  the  second  member  of  (23)  is  seen  to  have  the 
property  already  indicated. 

As  to  the  third  integral  in  the  second  member  of  (23),  let  us  write 


(25) 


f  /(tt  +  t)<p(n,  t)dt  =  f(ir  -  0)  J  <pin,  t)dt 


^-  j_  [fia+t)-fi7r-0)]<p{7i,t)dt, 


noting  that  /(tt  —  0)  necessarily  exists  since,  according  to  the  hypotheses  in 
Theorem  III,  the  function  f{x)  is  of  limited  total  fluctuation  in  the  neighborhood 
at  the  left  of  the  point  x  =  tt.  Upon  comparing  (25)  with  (16)  and  noting  the 
statements  in  §  46  connected  with  the  latter,  we  see  at  once  that  as  n  =  <x>  the 
expression  (25)  approaches  the  limit  |/(7r  —  0).  Likewise,  as  n  =  oo  the  last 
term  of  (23)  is  seen  to  approach  the  limit  |/(—  tt  +  0). 

In  case  x  =  —  ir  (instead  of  x  =  tt)  we  have  the  following  equations  corre- 
sponding to  (22)  and  (23) : 

Sn(-  Tr)  =  j  V(-  ^  +  t)<P(n,  t)dt  =(J    +   f'^)/(-vr+  tMn,  t)dt 

/(-  TT  +  t)ip{n,  t)dt  +  J    /(tt  +  t)<p{n,  t)dt 

or 

Sn{-  t)  =    f    /(tt  +  t)^{n,  t)dt  +    r  /(-  TT  +  tMn,  t)dt 
(26)  -"-  0  '  ^. 

+         f(7r-\-t)<p{n,t)dt+        f{-7r+t)cp{7i,t)dt, 

and,  upon  considering  the  four  integrals  here  appearing  on  the  right  as  we  con- 
sidered those  in  (23),  we  find 

lim  5„(-  tt)  =  0  +  0  +  i/(7r  +  0)  +  i/(-  tt  +  0). 


Formation  of  General  Theory  115 

Thus,  the  proof  of  Theorem  III  becomes  complete. 

49.  Theorem  IV  likewise  follows  from  (I),  (II)',  (III)  and  (IV)  upon  noting 
that  the  expressions 

^^-^  [So(±  X)  +  5i(±  7r)   + \-  Sn{±  X)] 

may  be  obtained  by  replacing  (p(n,  t)  by  ^(n,  t)  in  (23)  and  (26)  and  that,  as  a 
result  of  (IV),  we  have  ^{n,  t  dz  27r)  =  ^{n,  t). 

II 

The  Representation  of  Arbitrary  Functions  by  Means   of  Definite 

Integrals.    The  Formation  of  a  General  Theory  for  the  Study 

OF  the  Summability  and  Convergence  of  Fourier  Series  and 

Other  Allied  Developments 

50.  The  manner  in  which  the  summability  and  convergence  of  Fourier  series 
has  been  shown  in  §§  47-49  to  depend  upon  the  properties  of  the  integrals 

f<p{n,t)dt,        f^{n,t)dt, 

where  (p(n,  t)  and  $(n,  t)  are  defined  by  (5)  and  (19)  readily  suggests  the  general 
problem  of  determining  a  set  of  sufficient  conditions  for  any  function  ip{n,  t)  of 
the  two  real  variables  n,  t,  or  more  generally,  for  any  function  (p{n,  a,  t)  of  the 
three  real  variables  n,  a,  t  in  order  that  the  integral  (cf.  (6)) 


(27) 


fb—a  /^h 

f{a  +  i)<p{n,  a,  t)dt         or  I    f(x)<p(n,  a,  x  —  a)dx 

..  -a  'J a 


shall  converge  when  n  =  oo  to  the  values  \  [/(a  —  0)  +  /(a  +  ())]  or  \  [f(b  —  0) 
+  /(a  +  0)]  according  asa<Q;<6orQ;  =  either  a  or  b.  Naturally,  the  range 
of  possible  existence  for  such  functions  cp  will  depend  upon  the  conditions  im- 
posed upon  the  given  function  f{x)  when  considered  throughout  the  interval 
(a,  6),  and  in  determining  the  form  of  such  conditions  we  shall  hereafter  be 
guided  by  the  limitations  upon  f(x)  occurring  in  the  Theorems  of  §  46.  The 
general  theorems  about  to  be  obtained  will  serve  as  a  foundation  for  the  dis- 
cussion in  §§  64-70  relative  to  the  summability  and  convergence  of  the  well- 
known  developments  in  terms  of  Bessel  functions,  Legendre  functions,  etc, 

51.  Theorem  I.  Let  cpin,  a,  t)  be  a  function  of  the  real  variables  n,  a,  t  ichich, 
when  considered  for  values  of  a  lying  tvithin  any  sub-interval  {a',  b')  of  {a,  b)  (a  <  a' 
<  6'  <  &)  satisfies  the  following  three  relations  in  ichich  n  is  restricted  to  'positive 
integral  values  and  in  which  e  represents  a  positive  quantity  lohich  may  be  taken 
arbitrarily  small: 


(I)  lim   I     (p(w,  a,  t)dt  =  ] 


—  \     when    a  —  a  =  t^  —  €, 
\     when     e  ^  t  ^  b  —  a. 


116  SUMM ABILITY    OF   FOURIER   SeRIES    AND    AlLIED    DEVELOPMENTS 

Moreover,  let  these  limits  be  approached  uniformly  for  all  of  the  same  values  of 


a  and  t}^ 


(II)  ^{n,a,t)dt\<  A; 


where  A  represents  a  constant  independent  of  n,  a  and  t. 

(Ill)  I  (p{ii,  a,t)\<B;         a  —  a^t  ^  —  e     or     e  ^  t  ^  h  —  a, 

where  B  represents  a  constant  independent  of  n,  a  and  t. 

Aho,  letf{x)  be  any  function  satisfying  the  folloioing  two  conditions: 

{A)  When  considered  throughout  the  interval  {a,  b),f(x)  remains  finite  with  the 

possible  exception  of  a  finite  number  of  points  and  is  such  that  the  integral 


I     1/(0.-)  I  dx 

'J  a 


exists. 

(B)  When  considered  in  an  arbitrarily  small  neighborhood  about  the  (special) 
point  X  =  a  (a'  <  a  <  b')  f{x)  has  limited  total  fluctuation. 

Then  we  shall  have  for  the  {special)  value  of  a  mentioned  in  (B) 


(28) 


lim   r  f{x)<p{7i,  a,  x  -  a)dx  =  \  [/(«  -  0)  +  /(a  +  0)]. 

m— 00  ^ a 


Moreover,  if  {instead  of  condiiion  {B))  f{x)  is  continuous  throughout  the  interval 
{a',  b'),  the  points  x  =  a' ,  x  =  b'  included,  and  has  limited  total  fluctuation  through- 
out an  interval  (oi,  6i)  such  that  a  <  oi  <  a'  <  b'  <  bi  <  b,  then  tve  shall  have 
uniformly  for  all  values  of  a  in  {a',  b') 


(29) 


lim    1    f{x)(p{n,  a,  x  —  a)dx  =  f{a). 

7i  =  oa  t'a 


"  Thus,  to  an  arbitrarily  small  positive  quantity  a  it  shall  be  possible  to  determine  a  value  n, 
independent  of  both  a  and  I  such  that 


pt  I 

I    <p{n,  a,  t)dl  +  I    <  0-;        n  >  no- 


provided  a  and  I  are  assigned  values  consistently  with  the  relations 

a'  <  a.  <h';        a  —  w^f^  —  €. 
Likewise, 

1   /*'  I 

I     <p(n,  a,  t)dt  —  5     <  0-;         n  >  tie 
I «/  0  I 

provided  a  and  I  are  assigned  values  consistently  with  the  relations 

a'  <  a  <  b';  e'^t^b  —  a. 

It  may  be  added  that  in  case  one  confines  the  attention  to  the  convergence  of  the  integral 
(27)  for  special  values  of  a  (thus  not  considering  questions  of  uniform  convergence)  it  suffices 
that  relation  (I)  shall  be  satisfied  for  each  special  value  oi  a  {a'  <  a  <  b').  Similarly,  the  con- 
stants A  and  B  of  (II)  and  (III)  may  then  depend  upon  a. 


Formation  of  General  Theory  117 

Proof.— The  proof  of  this  theorem  is  readily  supplied  upon  referring  to  the 
methods  employed  in  §  46  for  the  study  of  the  integral  (6).  We  shall  therefore 
merely  indicate  the  essential  steps. 

Representing  the  integral  (27)  by  *„(«),  we  first  write  (cf.  (7)) 

m<p(7i,  a,  X  -  a)dx  +    1     f(x)<p(n,  a,  x  -  a)dx 

+  /      f(x)<p(n,  a,  x  -  a)dx  +  f(xMn,  a,  x  -  a)dx. 

Of  the  four  integrals  here  appearing,  the  first  two  approach  the  limit  zero 
asn=  CO  and  the  convergence  is  uniform  for  all  values  of  a  such  that  a'  <  a  <  b' 
as  results  from  (I),  (III)  and  (A).  Moreover,  the  third  and  fourth  integrals 
(considered  for  any  special  value  of  a  such  that  a'  <  a  <  h')  approach  respec- 
tively the  hmits  i/(«  -  0),  i/(a  +  0),  as  results  from  (I),  (II)  and  {B)  (cf.  (17), 
(18)). 

Likewise,  upon  comparison  with  the  corresponding  studies  in  §  46,  it  appears 
that  equation  (29)  will  hold  true  uniformly  under  the  conditions  stated  in  the 
theorem. 

52.  Theorem  II.  Let  ip{n,  a,  t)  he  a  function  of  the  real  variables  n,  a,  t 
which,  when  considered  for  values  of  a  such  that  a  <  a'  <  a  <  b'  <  b,  satisfies 
relations  (I)  and  (III)  o/  §  51  and  is  such  that  if  we  place 

(30)       ^{n,  a,  t)  =  ^-^-p^  {o{n,  a,  t)  +  ^{n  -  1,  a-,  0  +  •  •  •  +  ^(0,  a,  t)] 

the  following  relation  is  satisfied:  corresponding  to  a  given  e  >  0  ice  shall  have  for  all 
subsequenthj  chosen  sufficiently  large  values  of  n 

OT'  j    \Hn,a,t)\dt<  C 

where  C  represents  a  constant  independent  of  n,  a  and  e. 

Also,  let  fix)  be  any  function  which  satisfies  condition  (A)  of  §  51  together  with 
the  following : 

{BY  When  considered  in  the  neighborhood  of  the  {special)  point  x  =  a  (a'  <  a 
<  b'),  the  limits  f  {a  -  0),  f{a  +  0)  exist. 

Then  ive  shall  have  for  the  (special)  value  of  a  mentioned  in  (B)' 


(31) 


hm   I   f(x)^{n,  a,x-  a)dx  =  \  [f{a  -  0)  +  f{a  +  0)1. 


^  Moreover,  if  {instead  of  condition  {B)')  f{x)  is  continuous  throughout  the  interval 
(a',  b'),  the  points  x  =  a',  x  =  b'  included,  we  shall  have  uniformly  for  all  of  the 
same  values  of  a 


lim   I   f{x)^{n,  a,  x  -  a)dx  =  /(a). 


118  SmiMABILITY   OF   FoURIER   SERIES   AND   ALLIED    DEVELOPMENTS 

The  proof  of  this  theorem,  Hke  that  just  indicated  for  Theorem  I,  is  at  once 
suppHed  upon  following  the  steps  indicated  in  §  46  with  reference  to  the  special 
integral  (6)  there  occurring.     We  therefore  omit  it. 

53.  As  a  generalization  of  the  Theorem  III  of  §  46  we  have  the  following 
Theorem  III.     Let  (p(n,  a,  t)  he  a  function  of  the  real  variables  n,  a,  t  loliich, 
lohen  considered  for  the  special  values  a  =  a,  a  =  b  (b  >  a)  satisfies  the  following 
four  relations  in  ivhich  n  is  restricted  to  positive  integral  values  and  in  which  e  repre- 
sents a  positive  quantity  ichich  may  he  taken  arbitrarily  small: 


(Da.i 


lim   I    (p{7i,  a,  t)dt  =  —  |        when        a— b-\-€^t^  —  e, 

n=oo  t/0 

lim   J     (p(7i,  h,  t)dt  =  ^         when         e  '^  t  ^  6]—  a  —  e. 


(II)a,b  Relation  (II)  of  §  55  is  satisfied  ivhen  a  =  a  and  t  lies  in  the  interval 
0  ^  i  ^  e;  also  when  a  =  h  and  t  lies  in  the  interval  —  e  ^  t  ^  0. 

I  I  (p{n,  a,  t)\  <  B     ivhen     a—  b-\-e'^t^  —  e, 
^  1  !  <p{n,  b,t)\<  B     when     e  ^  t  ^  h  -  a  -  e, 

where  B  is  a  constant  independent  of  both  n  and  t. 

(IV)  <p{?i,  a,  t  -\-  b  —  a)  =  (pill,  b,t-\r  b  —  a)  =  <p{n,  a,  t)  =  (p{n,  b,  t). 

Also,  let  f{x)  he  any  function  which  satisfies  condition  (A)  of  ^  51  and  is  such 
that  in  arbitrarily  small  neighborhoods  to  the  right  of  the  point  x  =  a  and  to  the  left 
of  the  point  x  =  b  it  has  limited  total  fiuctuation. 

Then  we  shall  have 

lim  I    f(x)(p{n,  a,  x  —  a)dx  =  lim  I    f{x)(p{n,  b,  x  —  b)dx 

=  |[/(6-0)+/(a  +  0)]. 

Proof. — Here  again  the  proof  may  be  easily  supplied  upon  reference  to  the 
analysis  occurring  in  §  48.     Thus,  for  the  case  in  which  a  =  6  we  may  write 

Snih)  =    f  f{h  +  tMn,  b,  t)dt  =  (  f       Vr'+r)/(6  +  t)<p{n,  b,  t)dt, 

which,  upon  making  the  transformation  t'  =  b  —  a  -{-  tin  the  first  integral  of  the 
last  member  and  making  use  of  (IV)  becomes 

»0 


?n(&)  =    f        f{h  +  tMn,  b,  t)dt  +   f   /(6  +  t)<p{n,  b,  t)dt 


-b+( 

+   1    /(a  +  t)(p{n,  a,  t)dt. 
Jo 


Formation  of  General  Theory  119 

Of  the  three  integrals  here  appearing,  the  first  approaches  the  Hmit  zero  when 
n  =  00 ,  as  results  from  (1)^,  b,  (ni)a,  i  and  {A)  while  the  second  and  third  approach 
respectively  the  Hmits  |/(6  —  0)  and  |/(a  +  0),  as  results  from  (!)„,&,  (II) a,  b 
and  the  assumption  regarding  the  behavior  of  f{x)  in  neighborhoods  arbitrarily 
near  to  the  points  x  =  a  and  x  =  b. 

Similarly,  in  case  a  =  a  we  may  write 

Sn{a)  =  /(a  +  t)ip{n,  a,  t)dt  =  (        +  +  \j(^a-\-t)cp{n,  a,  t)dt 

»>'o  \  Jo  Je  Jt)~a-e  / 

Xb—a—e  /»0 

f{a  +  i)(p(n,  a,  t)dt  +  j    f{h  +tMn,h,  t)dt 

+   1    /(a  +  t)(p{7i,  a,  t)dt, 

Jo 

from  which  we  deduce  the  indicated  result  as  before. 

54.  Again,  we  have  (cf.  the  remarks  in  §  49  on  Theorem  IV)  the  following 
Theorem  IV.     Let  (p(n,  a,  t)  he  a  function  of  the  real  variables  n,  a,  t  which, 

when  considered  for  the  special  values  a  =  a  and  a  =  b  satisfies  relations  {T)a,b, 
(III)a,6  and  (IV)  of  §  54  and  also  the  following : 
The  integrals 

(II)'a,6  j    |$(n,  -  1,  t)\dt,        j     |$(n,  1,  t)\dt         (€  >  0), 

when  considered  for  all  values  of  n  sufitciently  large  remain  less  that  a  constant 
(independent  of  e). 

Also,  let  f(x)  be  any  function  which  satisfies  condition  (A)  of  §  51  and  is  such 
that  the  limits  f{a  +  0) ,  /(6  —  0)  exist. 

Then  we  shall  have 

X>>  pb 

f{x)^{n,  a,x  —  a)dx  =  hm  I    f{x)^{n,  b,  x  —  b)dx 

=  H/(&-0)+/(a+0)]. 

55.  Besides  the  relations  given  in  Theorems  III  and  IV  concerning  the  func- 
tions (p{n,  a,  t)  and  (p{n,  b,  t)  (which  relations  are  satisfied  in  particular  by  the 
function  (5)  pertaining  to  Fourier  series,  with  a  =  —  tt  or  a  =  tt)  it  is  important 
to  note  certain  others  which  we  shall  find  fulfilled  by  some  of  the  functions 
(p{n,  a,  t)  met  with  in  the  succeeding  pages  but  which  are  not  fulfilled  by  (5). 
These  relations  together  with  their  effects  upon  the  limiting  values  of  the  integrals 

I    f{x)(p{n,  a,  X  —  a)dx,  I    /(.r)$(?i,  a,  x  —  a)dx 

*Ja  Ja 

we  now  summarize  in  the  following  four  theorems : 


120  SUMMABILITY   OF  FOUBIEE  SeEIES   AND   AlLIED   DEVELOPMENTS 

Theorem  V.  Let  (p(n,  a,  t)  he  a  function  of  the  real  variables  n,  a,  t  which, 
when  considered  for  the  special  value  a  =  a  satisfies  the  following  three  relations 
in  which  n  is  restricted  to  positive  integral  values  and  in  ichich  e  represents  a  positive 
quantity  which  may  he  taken  arbitrarily  small: 

(I)a  lim   I     <p(n,  a,  t)dt  =  Gi;         e  ^t  ^h  —  a         (b  >  a), 

Gi  being  a  constant  (independent  of  t) . 

(II) a  Relation  (II)  of  §  51  is  satisfied  when  a  =  a  and  0  ^  i  ^  e. 

(Ill)a  \<p{n,  a,t)\<  B;         e  ^  t  ^  b  —  a, 

B  being  a  constant  independent  of  n  and  t. 

Also,  let  f(x)  be  any  function  ivhich  satisfies  condition  {A)  of  §  51  and  is  such 
that  it  has  limited  total  fluctuation  in  an  arbitrarily  small  neighborhood  at  the  right 
of  the  point  x  =  a. 

Then  ice  shall  have 

lim    I    f(x)(p{n,  a,  X  —  a)dx  =  Gif{a  +  0). 

,  n=oo  tJa 

Theorem  VI.  Let  (p{n,  a,  t)  be  a  function  of  the  real  variables  n,  a,  t  ichich, 
when  considered  for  the  special  value  a  =  h  satisfies  the  following  three  relations  in 
which  n  is  restricted  to  positive  integral  values  and  in  which  e  represents  a  positive 
quantity  which  may  be  taken  arbitrarily  small: 

(1)6  lim    I    (p{n,  b,  t)dt  =  —  G2;         a—  b^t^  —  e         (b  >  a), 

G2  being  a  constant  (independent  of  t) . 

(11)6  Relation  (II)  of  §  51  is  satisfied  when  a  =  b  and  —  e  ^  t  ^  0. 

(111)6  \<p(7i,  b,t)\<  B;         a-  b^t^  -  €, 

B  being  a  constant  independent  of  both  n  and  t. 

Also,  let  f(x)  be  any  function  which  satisfies  condition  (A)  of  §  51  and  is  such 
that  it  has  limited  total  fluctuation  in  an  arbitrarily  small  neighborhood  at  the  left 
of  the  point  x  =  b. 

Then  we  shall  have 

lim    r  f(x)ip(n,  b,  x  -  b)dx  =  G2f(b  -  0). 

71=30     *J  a 

Theorem  VII.  Let  (p(n,  a,  t)  be  a  function  satisfying  relations  (I)a  and  (Ill)a 
of  Theorem  V  but,  instead  of  (II)a,  the  following : 

(11)  a  Relation  (II)'  0/  §  54  i^  satisfied  when  a  =  a,  it  being  understood  that 
the  integration  there  appearing  is  then  taken  from  0  to  e  instead  of  from  —  e  to  e. 


Formation  of  General  Theory  121 

Also,  let  f{x)  be  any  function  which  satisfies  condition  {A)  of  §  51  aiid  is  such 
that  the  limit  f  (a  +  0)  exists. 
Then  ive  shall  have 

lim    I    f(x)^{n,  a,  X  —  a)dx  =  Gif(a  +  0), 

n=co  >J  a 

where  <J>  is  defined  by  (30) . 

Theorem  VIII.  Let  (p{n,  a,  t)  be  a  function  satisfying  relations  (X)b  and 
(111)6  of  Theorem  VI  but,  instead  of  (II)^,  the  following : 

{ll)b  Relation  (II)'  of  §  52  is  satisfied  when  a  =  b,  it  being  understood  that  the 
integration  there  appearing  is  then  taken  from  —  e  to  0  instead  of  from  —  e  to  e. 

Also,  let  f{x)  be  any  function  ivhich  satisfies  condition  {A)  of  §  51  and  is  such 
that  the  limit  f(b  —  0)  exists. 

Then  tve  shall  have 

lim   r  fixMn,  b,  x  -  b)dx  =  6^2/(6  -  0), 

n=oo  tJa 

where  <J>  is  defined  by  (30) . 

The  first  of  the  Theorems  V,  VI  results  directly  upon  writing 

Xb—a  /     /"e  nb—a  \ 

f{a  +  t)ip{ii,  a,  t)dt  =  (    I     +    1         )/(«+  t)^{n,  a,  t)dt;         e  >  0 

and  then  applying  to  each  of  the  last  two  integrals  the  methods  already  used 
in  §  48  for  the  study  of  similar  integrals. 
Theorem  VI  likewise  results  upon  writing 

(32)     Snib)  =    r   /(6  +  tMn,  b,  t)dt  =(  f  +   f  ^  )f{b  +  tM7i,  b,  t)dt. 

Ja-b  \  J-e  Ja-b    J 

The  proofs  of  Theorems  VII  and  VIII  being  likewise  readily  supplied,  are 
suppressed. 

56.  We  proceed  to  make  certain  observations  which  will  prove  useful  in 
applying  the  general  theorems  of  §§  51-55  to  special  integrals  (27). 

(1)  If  in  applying  Theorem  I  of  §  51  it  is  found  that  for  some  special  value 
of  t  different  from  zero,  ^  =  ^1  4=  0  say,  the  function  (p{n,  a,  t)  becomes  infinite 
or  otherwise  is  of  such  a  character  that  uncertainty  arises  concerning  any  one 
of  the  relations  (I),  (II),  (III)  when  t  =  t\,  then  the  theorem  will  still  hold  good 
provided  that  it  can  be  shown  that  the  integral 

h=    {        \f{a+t)<p{:n,a,t)\dt, 

where  ^  is  arbitrarily  small  and  >  0,  approaches  (n  =  00 )  uniformly  the  limit 
zero  for  a  <  a'  ^  a  ^  6'  <  6,  or  else  is  such  that  for  the  same  values  of  a  and 


122  SUMMABILITY   OF   FoURIER   SeRIES   AND   ALLIED    DEVELOPMENTS 

for  all  (positive  integral)  values  of  n  the  same  integral  approaches  uniformly 
the  limit  zero  as  ^  =  0. 

An  examination  of  the  method  used  in  proving  Theorem  I  shows  at  once  the 
correctness  of  this  remark.  More  generally,  in  case  of  uncertainty  of  any  kind 
in  the  behavior  of  f{a  +  t)(p{n,  a,  t)  for  the  value  t  =  ti  ^  0  (a  —  a<ti  <b  —  a), 
it  suffices  for  the  existence  of  (28)  and  (29)  that  relations  (I),  (II),  (III),  (A) 
and  (B)  (or  in  (29)  the  substitute  for  (B)  there  mentioned)  shall  be  satisfied 
throughout  the  two  intervals  (a  —  a  ^  t  ^  ti  —  ^),  {h  -\-  ^  ^  t  ^  b  —  a) 
(^  arbitrarily  small  and  positive)  instead  of  throughout  the  whole  interval  (a  —  a, 
h  —  a),  provided  merely  that  the  expression  I^  above  defined  has  either  of  the 
properties  just  mentioned. 

If  the  exceptional  point  h  h  =  a  —  a  then,  instead  of  the  two  intervals,  we 
have  to  consider  the  single  one  {a  —  a-\-^-^t^h  —  a),  while  instead  of  I^ 
as  defined  above,  we  shall  have  to  consider  the  integral 

h^"^  =    I      "      \Koi+t)<p{n,a,t)\dL 


:-)=    r  "      \f{a-^t)^{n,a,t) 

'J  a— a 


A  corresponding  statement  may  at  once  be  supplied  for  the  case  in  which  the 
exceptional  point  is  ifi  =  h  —  a. 

In  the  case  of  two  or  more  of  the  exceptional  points  h  {a  —  a  ^  h-^h  —  a) 
the  corresponding  statements  are  readily  supplied. 

(2)  The  conditions  demanded  in  Theorem  II  may  be  stated  without  reference 
to  the  function  (p{n,  a,  t).  Thus,  it  suffices  (aside  from  the  conditions  upon 
f{x))  that  the  function  <l>(n,  a,  t)  shall  satisfy  relations  (I)  and  (III)  of  Theorem  I 
together  with  (II)'  of  Theorem  II. 

This  follows  from  the  fact  that  the  conditions  placed  upon  (p{n,  a,  t)  in 
Theorem  II  are  there  inserted  merely  that  $(w,  a,  t)  may  have  the  properties 
just  indicated,  the  latter  being  those  upon  which  the  proof  in  reality  depends. 

Similarly,  in  using  Theorems  VII  and  VIII  the  conditions  stated  relative  to 
(p{n,  a,  t),  (p{n,  b,  t)  may  be  replaced  by  the  same  conditions  referred  to  ^{n,  a,  t), 
$(n,  b,  t). 

(3)  Assuming  that  relations  (II),  (III),  {A)  and  (J5)  of  Theorem  I  are  satis- 
fied, let  us  suppose  that  instead  of  relation  (I)  we  have  the  following  :^^ 


(I)'        lim   I    ip{n,  a,  t)dt  = 

n=oo  c/q  I 


~"  2  H~  x(oi,  t)     ivhen     a  —  a  ^  t  ^  — 
h  +  x(aj  0     when     e  ^  t  ^  b  —  a, 


where  x(a,  t)  is  any  function  of  a  and  t  such  that 

(a)  Having  given  an  arbitrarily  small  positive  quantity  a,  one  may  determine 

a  positive  quantity  ^  dependent  only  upon  a  such  that 

"  As  in  (I)  of  §  51,  it  is  here  to  be  understood  that  the  convergence  (n  =  oo )  is  uniform 
for  the  indicated  values  of  a  and  /. 


Formation  of  General  Theory  123 

a'  ^  a  ^  h', 


xia,  t)\<  a     when      i  _  t  <  /  <  t 

(6)  The  partial   derivative  dxfdt  exists  whenever  a'  ^  a  ^  b',   a  —  a  '^  t 
b  —  a  and  for  the  same  vakies  of  a  and  t  is  such  that 


dx 
dt 


<  D  =  constant  independent  of  a  and  t. 


Under  these  conditions  it  is  easily  seen  that  the  function  (p{n,  a,  t)  —  dxl^t 
comes  to  satisfy  relations  (I),  (II)  and  (III)  of  the  theorem  of  §  51  from  which 
it  follows  that  for  a  fixed  value  of  a  such  that  a'  ^  a  <  6'  we  may  write 

-    I    /(^)     -^1  dx  +  hm  I    f(x)(p(n,  a,  x  -  a)dx  =  - S^^ • 

tJa  L  '-'^  J(=x— a  n=co  *Ja  ^ 

Moreover,  if  (instead  of  condition  (B))  f{x)  is  continuous  throughout  the 
interval  a'  ^  x  ^  b' ,  the  end  points  x  =  a',  x  =  b'  included,  and  has  hmited 
total  fluctuation  throughout  an  interval  {a\,  bi)  such  that  a  <  ai  <  a'  <  b' 
<  bi  <  b,  then  for  all  values  of  a  in  (a',  b')  the  equation  will  hold  true  uni- 
formly, it  being  understood  that  the  right  member  is  then  replaced  by /(a). 

Analogous  remarks  relative  to  Theorems  (III),  (V),  (VI)  are  readily  supplied. 


Ill 

The  Calculus  of  Residues  as  Applied  to  the  Series  Developments  for 
AN  Arbitrary  Function.^^    The  General  Problem  of  Sturm 

57.  A  comparison  of  the  developments  occurring  in  mathematical  physics 
for  a  function /(.t)  of  one  real  variable  x  shows  that  they  are  ordinarily  of  the  form 


(33) 


J  f{x)F{x)Hi{\n,  x)dx  J  f{x)Fix)H2{\n,  x)dx 

Ih{\n,  X)  -^^h 1-   H2(\n,  X)  ~^^ 

F(x)Hi'(Kn,x)dx  F{x)H2\\n,  x)dx 

I   f(x)Fix)Hm{\n,  x)d:i 

*J  a 

j'  F(x)HJ{\n,  x)dx 


+  •  •  •  +  i/„.(X«,  x) 


where  //i(X„,  x),  H2(kn,  x),  •  •  •,  HmO^n,  x)  are  m  functions  of  x  and  of  a  certain 
parameter  X  which  takes  different  values  from  term  to  term  in  (33)  according  to 
some  given  law,  and  where  F{x)  is  a  function  of  x  only  which  is  finite  throughout 
the  interval  (a,  b). 

Thus  in  the  case  of  a  Fourier  series  we  have  m  =  2,  i/i(X„,  x)  =  sin  nx, 
-^2(Xn,  x)  =  cos  nx;  and  a  =  —  t,  b  =  ir,  F{x)  =  1.  Again,  in  deahng  with  the 
usual  expansion  of  f{x)  in  terms  of  Bessel's  function  of  order  zero,  we  have 
m  =  1,  //i(Xn,  x)  =  Jo(Xn,  x),  a  =  0,  b  =  1,  F{x)  =  x,  Xn  being  one  of  the  roots 
of  the  transcendental  equation  Jo{x)  =  0. 

It  is  to  the  important  developments  (33)  that  we  shall  hereafter  devote  our 
attention. 

The  first  n  terms  of  (33)  when  considered  for  any  particular  value  of  x  such 
as  X  =  a  may  evidently  be  put  into  the  form 


b 

f{x)<p{7i,  a,  X  -  a)dx, 


where 


1^  The  calculus  of  residues  was  first  applied  by  Cauchy  to  the  study  of  infinite  series,  in 
particular  to  Fourier  series  (cf.  Picard,  "Traite  d' Analyse,"  Vol.  II,  Chap.  VI,  §  9  e<  seq).  Its 
application  to  the  general  study  of  developments  in  terms  of  normal  functions  appears  to  have 
been  first  made  by  Dini  (cf.  "Serie  di  Fourier,  etc.,"  §§61-64)  upon  whose  investigations  the 
present  §  is  based. 

124 


Calculus  of  Residues  125 

(34)  (p{n,  a,  X  —  a)  =  2^  2^  //s(Xr,  a)  ~t, • 

^=''='  I    F(x)Hs\Kx)dx 

Upon  referring  to  the  theorems  of  §§  51-55  it  thus  appears  that  in  order  to 
show  the  summabihty  or  convergence  of  series  (33)  to  the  value 

/(a-0)+/(^+0)  /(a+0)+/(&-0)  r-  /^    a_  m  r  ^f^      m 

according  to  the  cases  there  considered  it  suflSces  to  show  that  the  conditions 
specified  for  (p{n,  a,  t)  in  the  same  theorems  are  present  when 


(35)  ip{n,  a,t)  =  2^2^  Hs{\r,  a)  — —^ . 

"='■'='  I    F{t)H/CKr,t)dt 

Thus  the  integral 

(36)  I    ^(w,  a,  /)(Z^, 

Jo 

which  plays  an  important  part  in  these  theorems,  becomes  in  the  present  case 

^t  n       r.  f    F{a  +  t)H,(Kr,  «  +   t)dt 

(37)  I    (p(n,  a,t)dt  =  J^J2  Hs{\r,  a)  — zr, . 

^"  '=''^'  j    F{t)Hs'{Ki)dt 

58.  Now,  the  values  Xi,  X2,  X3,  •  •  • ,  Xn,  •  •  •  of  the  parameter  X  are  usually 
given  as  the  roots  (or  part  of  the  roots)  of  some  transcendental  equation  ^(2)  =  0 
where  u(z)  is  a  function  of  the  complex  variable  z  which  is  analytic  throughout 
all  finite  portions  of  the  z  plane.  Thus  in  the  case  of  Fourier's  series  we  have 
u{z)  =  sin  TTS  and  in  the  above  mentioned  case  of  the  expansion  in  Bessel's 
function  of  order  zero  we  have  u{z)  =  Jo{z).  Moreover,  these  roots,  when 
considered  as  zeros  of  the  function  u{z)  are  ordinarily  zeros  of  the  first  order  and 
we  shall  suppose  this  to  be  the  case  in  what  follows. 

Then  the  function  iv(z)  =  l/u{z)  will  be  analytic  throughout  the  finite  z 
plane  with  the  exception  of  the  points  Xi,  X2,  X3,  •  •  • ,  X„,  •  •  • ,  where  it  will  have 
poles  of  the  first  order  and,  considering  d{z)  to  be  any  other  function  of  z  which 
is  analytic  throughout  the  finite  z-plane,  we  shall  have,  provided  p  is  a  positive 
integer, 

e{z)lV^(z){z  -  \nV  =   e{\n)AP  +  [d  (z)  10^^  (z)  {z  -  KVV.Sz  -  X,.)  +    •  •  • 

(38)  ^  [a(.)..(.)(.-X„).r'  (^  _  ,^)^.  ^  ^,_(^)(^  _  ,^),, 


126  SUIMMABILITY   OF   FoURIER   SERIES   AND   ALLIED    DEVELOPMENTS 

where  A  is  the  limit  of  w(s)(z  —  X„)  as  s  =  X„  and  ^1(2;)  is  a  function  of  z  which 
is  analytic  in  the  neighborhood  of  the  point  2  =  X„  and  where 

[d{z)w^(z){z-\ny]l 

indicates  the  value  of  the  5th  derivative  of  6{z)w^{z){z  —  \n)^  at  the  point  z  =  Xn- 
From  (38)  we  have 

e{z)^i-iz)  =  ^^^^^^  +  ■ ^^zr^^.       +  •  •  • 

(39) 

,  [e(z)iv^(z){z-\nyY.:'  ,     ,, 
+    (,_x„)(p-^yr"  +  ^'^^^^)' 

and  integrating  in  the  positive  direction  about  any  closed  contour  C  which 
encloses  the  point  2  =  X„  but  no  other  pole  of  w{z)  we  have  by  Cauchy's  integral 
theorem 

(^"'  2^  J^eW«-''fe)rf.  = (^^^lyi . 

If,  therefore,  we  integrate  about  a  closed  contour  Cn  which  encloses  the  first  n 
of  the  points  Xi,  X2,  X3,  •  •  •  but  no  other  poles  of  ^(2)  we  shall  have 

and  hence  also 

^*2^  l'i22sX.''(^^"'(^)''^  =  S (f^Tv. 

whenever  either  side  of  the  same  relation  has  a  meaning. 
In  particular,  when  p  =  1  and  p  =  2  we  have  respectively 

(43)  ^    f  d(z)lLiz)dz  =   T  [d{zMz)iz  -  Xn)]A„, 

(44)  ^.  f  d{z)w\z)dz  =  E  [e{z)w\z){z  -  X„)2] 

l-Kl  Jc„  n=l 

or,  since 
w{z){z  —  X„)  = 


Ki> 


u{z) 

2  —  X„ 


[e{z)w\z){z  -  \n)%  =    e\\n)[l0\\n){z  -  X„)^]  +  e{K)[w'{z)(z  -  ^n)']',. 

W'(X„)2  u'Oinf       ' 


Calculus  of  Residues  127 

relations  (43)  and  (44)  may  be  written  in  the  form 

(45)  ±.rm,,^±eiKi 

2inJc„u{z)  „=iw(X„)' 

^^^^  2-KiJcyiz)'^^  ntlU'(Xn)  u' (KnY         \' 

It  is  desirable  to  note  also  that  if  in  (46)  we  substitute  d{z)^{z)  for  ^(2)  we 
obtain 

so  that  if  \l/' (Kn)u' (hn)  —  \p(XnW{Xi)  =  0  we  shall  have 

eWnrnXn) 


2*'(X.)^ 


59.  We  proceed  to  apply  the  results  in  (45)  and  (47)  to  the  sum  (37)  which 
defines  the  integral  (36)  whose  properties  are  desired  in  order  to  investigate  the 
convergence  of  (33). 

Let  us  suppose  that  for  the  given  value  of  a  we  can  construct  a  function 
6{z)  which  shall  be  analytic  throughout  the  finite  2  plane  and  such  that  its  value 
at  the  points  Xi,  X2,  •  •  • ,  X„,  •  •  •  shall  be  given  by  the  equation 

^  //,(Xn,  a)   f  F(a  +  t)H,(Kn,  a  +  t)dt 

(48)  OiK)  =  E 'y^ u'(K). 

As  a  result  of  (45)  we  shall  then  have 

(49)  r  <p{n,  a,  t)dt  =  ^r~.  \    -~dz. 
Jo  2TriJc^u{z) 

If,  again,  we  can  construct  a  function  d{z)  analytic  throughout  the  finite  plane 
and  subject  to  the  single  restriction 

^  IIs{\n,  a)   f  F{a  +  t)IIs{K,  a  +  t)dt     ■ 

(50)  d'iK)  =  E ^^^-^ u'(Kny, 

iA(X„)        F(t)Hs'(Kn,t)dt 

where  \p{z)  is  any  function  of  z  analytic  throughout  the  finite  z  plane  and  such 
that  ^'(\n)u'0^n)  =  ^{K)u"{\n)  then,  upon  applying  (47)  we  shall  have 


(51)  J    <p{n,a,t)dt  =  ^^.j 


e(z)Hz)  . 
dz. 


i«  Cf.  Chapter  I,  formula  (30). 


128  SUMMABILITY   OF   FOUIIIER   SeRIES   AND   AlLIED   DEVELOPMENTS 

It  thus  appears  that  by  means  of  (49)  and  (51)^^  the  discussion  of  (37)  may 
sometimes  be  transferred  to  that  of  an  integral  of  a  complex  variable  z.  This 
will  be  the  case  in  the  special  developments  to  be  considered  in  what  follows. 

60.  We  now  proceed  to  examine  the  series  (33)  in  some  of  its  more  important 
cases — viz.,  those  related  to  the  general  problem  of  Sturm.^^  Here  we  have 
TO  =  1  and,  representing  by  //(X„,  x)  the  single  function  Hi{\n,  a*),  we  have  by 
hypothesis 

(52)  I    F{x)Hi\n,  x)H(Kn,  x)dx  =  0     tchen    n  ^  m. 

Moreover,  when  x  is  taken  between  a  and  h  (a  and  6  included)  the  function 
II{z,  x)  is  assumed  to  be  analytic  in  z  throughout  the  finite  z  plane  and  real  when 
z  is  real;  also  to  be  such  that  when  z  has  any  one  of  the  values  Xi,  X2,  •  •  •,  Xn,  •  •  • 
it  is  a  solution  of  a  certain  linear  differential  equation  of  the  form 

(53)  a^(^(^)^^r^)  +  iP(^>(^)  +  F,{x)]H{z,  X)  =  0, 

where  K(x),  F(x)  and  Fi{x)  are  functions  of  x  only,  while  v{z)  is  a  function  of  z 
only. 

In  such  cases  the  developments  (33)  assume  the  form 

(54)  JlqnH{\n,x), 

71  =  1 

where 


(55)  Qn  = 


r  f{x)F{x)H{\n,  x)d:^ 

f    F{xW(\n,  X)dx 


We  first  proceed  to  note  certain  general  consequences  which  flow  from  the 
above  restrictions  upon  H{z,  x). 
From  (53)  we  have 

(56)  ^(^(•'^)^^— )  +  {^(•^)KXJ  +  F^{x)]H(K,  X)  =  0, 

1^  It  is  to  be  observed  that  if  in  (49)  the  function  d{z)  has  singular  points  within  C„  the  formula 
continues  true  provided  that  the  sum  of  the  residues  of  the  right  integrand  at  such  points  be 
subtracted  from  the  second  member.  Similar  remarks  evidently  apply  in  (51)  if  0{z)i(/{z)  has 
singular  points  within  C„. 

1*  Cf.  DiNi,  "  Serie  di  Fourier,  etc.,"  §§  90-96.  The  problem  here  presented  has  been  the 
subject  of  numerous  and  extensive  researches  in  recent  years,  but  usually  under  the  assumption 
(not  here  introduced)  that  the  differential  equation  (53)  in  terms  of  whose  solutions  the  proposed 
development  is  to  be  made,  shall  have  no  singular  points  within  the  (closed)  interval  (a,  b)  for 
which  the  same  development  is  to  hold.  But  this  assumption  unfortunately  rules  out  some  of 
the  most  important  special  developments,  such  as  those  in  terms  of  Bessel  functions  and  Legendre 
functions.  For  summary'  remarks  upon  the  more  recent  researches  of  this  character,  see  Bocher's 
address  before  the  International  Congress  of  Mathematicians  at  Cambridge  in  August,  1912,  §  11, 


Problem  of  Sturm  129 

(57)  ^(^(•^)^^^^)  +  {F(^>0<n)  +  F,{x)]H{K,  X)  =  0. 

Hence,  after  multiplying  both  members  of  (56)  by  H(Kn,  x)  and  both  members 
of  (57)  by  H(\n,  x)  and  subtracting,  we  obtain 

F(X)  {V(\n)   -   v(Km)}HiXn,  x)H{\n,  x) 

r5S^  ^   I  z-r  \[  ur\        ^  dH(Kn,  x)  dH(\n,  x) "[  ] 

(58)  =  -  I  Kix)  |_Zf(X„,  X)  —^ F(X.,  X)  — ^^- J  I 

and  therefore 

dH(\m,  x) 


f  Fix) H (Km,  x)Hi\n,  x)dx  =    ,..^     ,.    .  I  Kix)  \Hi\n,  x) 


dx 


H{Ki,  X)  - 


ri]  ■ 


Thus,  in  order  that  (52)  may  be  satisfied  it  suffices  that  the  roots  Xi,  X2,  X3, 
be  so  chosen  that 


(59) 


K{b)    II(kn,  x) ^ H(\n,  x) ^ 


IV  \[  ur\        >  ^-^'(X^,  x)                       dH(Kn,  x)  1 
-  A  (a)     H(Kn,  x) ^ H(Km,  x) — =  0, 


provided  m  4=  w-  Moreover,  among  the  different  ways  in  which  this  relation 
may  exist  is  that  of  supposing  that  for  every  value  of  n  we  have  the  following 
two  equations  simultaneously: 


(60) 


K{x) ^ h'H(Kn,  x)  =  0    when    x  —  a, 

K{x) -~^ hH{\n,  x)  =  0     iDhen     x  =  b, 


h  and  h'  being  any  real  constants,  including  the  limiting  values  A  =  ±  °o , 
h'  =  dz  ^  corresponding  to  which  the  same  equations  become  //(X„,  a)  =  0 
and  //(Xn,  6)  =  0  respectively.  We  shall  hereafter  confine  our  attention  to  the 
cases  in  which  relations  (60)  are  satisfied.  Furthermore,  if  K(a)  =#  0,  K{b)  4=  0, 
we  shall  suppose  that  the  transcendental  equation  v{z)  =  0  whose  roots  deter- 
mine the  quantities  Xi,  X2,  X3,  •  •  •  is  taken  in  the  one  or  the  other  of  the  two 
following  manners: 

(61)  u{z)  =  [/i(.T)  -f^  -  h'Hiz,  X)  ]^  =  0, 

(62)  u{z)  =  [/v(a')  ^"^  -  hlliz,  x)  ^   =  0, 
10 


130  SUMMABILITY   OF   FOXJUIEE   SERIES   AND   AlLIED    DEVELOPMENTS 

thus  rendering  one  of  the  two  relations  (60)  satisfied  at  once.  Similarly,  if 
K(a)  =  0,  K(b)  4=  0  we  shall  use  (62).  In  this  case  it  is  to  be  observed  that  we 
have  merely  to  place  h'  =  0  (u{z)  having  been  chosen  as  indicated)  to  have 
equations  (60)  satisfied  ichatever  the  solution  H{z,  x)  of  (53)  chosen  to  be  used 
in  (54).  Likewise,  when  K{a)  =}=  0,  K{b)  =  0  we  shall  use  (61)  in  which  case 
the  solution  II{z,  x)  of  (53)  to  be  used  in  (54)  may  be  chosen  arbitrarily.  Finally, 
if  K{a)  =  0,  K{b)  =  0  the  equation  u{z)  =  0  may  be  taken  arbitrarily  together 
with  the  solution  II {z,  x)  without  destroying  the  coexistence  of  (60). 

61.  We  add  that  if  the  solution  II{z,  x)  considered  as  a  function  of  the  two 
variables  z  and  x  is  finite  and  continuous  together  with  its  first  and  second  partial 
derivatives :  dH/dx,  dH/dz,  d'^H/dxdz  for  all  real  values  of  x  such  that  a  ^  x  ^h 
and  for  complex  values  of  z  in  the  neighborhood  of  each  of  the  points  z  =  X„, 
and  if  the  equation  (61)  {h'  finite  or  infinite)  is  satisfied  identically  for  all  values 
of  2  in  these  regions,  then  it  is  easy  to  evaluate  each  of  the  integrals 

(63)  f  F{x)H\\n,  x)dx, 

which  appear  in  the  coefficients  g„  of  the  series  (54). 

In  fact,  if  we  change  \m  to  z,  as  we  may  now  do,  and  integrate  from  a  to  a; 
(a  <  X  <  h)  we  shall  have  by  (58)  and  (61) 


J^Fix)H(z,  x)H{\n,  x)dx  =  ^^^^  }  ^^^^  ^K{x)  [h{z,  x) 


dll(\n,  X) 


dx 

dl 
dx 


HQ^n,  x)  — ^^ —  j  J  , 


and  this  holds  true  for  any  value  of  z  in  the  indicated  regions. 

Whence,  upon  allowing  z  to  approach  the  value  Xn  we  obtain  under  the 
present  hypotheses 

\dH{\n,x)dH(\n,x) 


£F(xWiK,x)dx  =  ^^[Kix){ 


d\n  dx 

-  //(Xn,  X) 


d\ndx        J  J  ' 


where  if  desired  X„  may  be  changed  to  z  for  values  of  z  in  the  indicated  regions. 
Passing  now  to  the  limit  as  a:  =  6  we  obtain 


in  which  as  above  we  may  replace  Xn  by  z  provided  z  has  values  in  the  indicated 
regions. 


Pkoblem  of  Sturm  131 

Finally,  by  use  of  the  second  of  equations  (60)  we  may  write  (64)  in  the 
following  form  when  h  is  finite : 

(65)  £n^)u^,^^,^),.-^[m^.,^)\k'^^-^ -^w^l^^ll- 


(66) 


In  like  manner,  if  A  =  ±  co  so  that  i/(Xn,  6)  =  0  then  (64)  may  be  written 

dH{\n,  x)  dH{\n,  x) 


£  F(x)H'{\n,  x)dx  =  -^,^^^^^  ^^Kix) 


d\n  dx 


62.  Expressions  (64),  (65)  and  (66)  thus  enable  us  to  find  under  special  con- 
ditions the  value  of  the  integral  (63).  Among  the  cases  in  which  the  same 
special  conditions  cannot  be  satisfied,  the  following  are  to  be  especially  noted. 

If,  as  we  have  supposed,  F(x)  and  //(X„,  x)  are  real  when  x  is  such  that 
a  ^  X  ^  b  and  if  in  this  interval  F{x)  does  not  change  sign,  then  the  integral 
(63)  cannot  be  equal  to  zero.  Whence,  under  these  conditions  (64)  cannot  be 
used  if  K{b)  =  0  {h  finite  or  infinite)  or  if 

(67)  H(Kn,b)  =  0        {h  finite) 

or  if 


(68) 


veH(K,x)i^  remx.,x)^ 

L       5X„       J,      ^     o*^     L       dx        i      ^         {iitnjimte) 


or  (as  appears  from  (65))  if 

(69)  L  ^-axr^  -  ^^-""^  -^Kdx^  i  =  0        ^^'fi'''^'^' 

63.  Returning  then  to  the  series  (54)  and  assuming  that  the  quantities 
Xi,  X2,  X3,  •  •  •  are  taken  as  the  positive  roots  of  the  equation  (62)  while  the  equa- 
tion (61)  shall  be  satisfied  identically  for  all  values  (real  or  complex)  of  z  in  the 
neighborhoods  of  the  same  values;  assuming  also  that  the  partial  derivatives  of 
H{z,  x)  exist  and  satisfy  such  other  conditions  as  we  have  imposed  in  §  61,  we 
may  say  that  unless  K{b)  =  0  or  one  of  the  conditions  (67),  (68)  or  (69)  is  satis- 
fied, we  shall  have  for  such  developments  when  h  is  finite 


uiz)  =  [k{x)^--^^  -  hlliz,  x)'^^, 


J  F(x)IP{\n,  x)dx  =  -^-^  [/i(X„,  .r)  I  a' 


dllO^n,  X) 
din 

(70) 


-^(^)-ax^„aV--|Jr"7(>o''^'-'^- 

On  the  other  hand,  if  A  =  ±  <» ,  we  shall  have 

u{z)  =  II(z,  b), 


-   Rn, 


132  SUMMABILITY   OF   FOURIER   SERIES   AKD   ALLIED    DEVELOPMENTS 

(71)  f  n.yHKK,  .)</.  =  "^  [a'W^-^^^]^. 

upon  applying  formulas  (49)  and  (51)  we  thus  obtain  the  following  general 
results  concerning  the  integrals  (36)  pertaining  to  the  present  developments: 
(1)  h  finite.     Formula  (49)  here  gives 

^     ^       i^iz)H{z,a)j^   F{a+t)H{z,a-\-t)dt 

(72)  I     <p{7i,  a,  i)dt  =  TTT-  \ '~T~^dH  1 

where  Rn  represents  the  sum  of  the  residues  of  the  integrand  at  any  singular  points 
which  it  may  have  within  C„  besides  the  points  Xi,  X2,  •  •  • ,  Xnl  i-  e.,  besides  those 
points  z  =  \n  within  C„,  for  which 

(73)  ^K{x)^^-hH^^^u{z)  =0. 

Formula  (51)  here  gives 

r  ,  1     r        e{z)yp{z)dz  ^ 

(74)  J^    <p{n,  a,  t)dt  =  ^.  J^^  f     a^_        T  "  ^' 

L      dx         '     J, 

where  R„  represents  the  sum  of  the  residues  of  the  integrand  at  any  singular 
points  which  it  may  have  within  C„  besides  those  points  2  =  Xn  for  which  (73) 
exists,  where  \l/(z)  is  a  function  of  z  only  such  that  \}/'(KnWO^n)  —  '/'(^n)w"(Xn)  =  0 
and  where  6{z)  is  to  be  so  determined  that 

v'{\n)u'{\n)H(Kn,  a)  J    F{a  +  t)II{Xn,  a  +  t)dt 

(75)  e'iK)  =  ^kkWo^)  ■ 

(2)  h  =  ±  ^ .     Formula  (49)  here  gives 

^      .  p'{z)Hiz,  a)  J  F{a  +  t)n{z,  a  +  t)dt 

(76)       I    <p{n,  a,  t)dt  =  ^  I    /    \jjx dz  -  R^, 

•^«  ^^'^^»  (^Kj^JH(z,b) 

where  Rn  represents  the  sum  of  the  residues  of  the  integrand  at  any  singular 
points  which  it  may  have  within  C„  besides  those  points  for  which 

H{z,  b)  =  u{z)  =  0. 
Formula  (51)  here  gives 


r   f       A^/      ^    r<P(^)e(z)dz 


Problem  of  Sturm  I33 

where  R^  represents  the  sum  of  the  residues  of  the  integrand  at  any  singular 
points  which  it  may  have  within  C„  besides  those  points  for  which 

H(z,  b)  ^  u{z)  =  0, 
where 

and  where 

v'0^n)u\■Kr^)H(K,  a)   f  F{a  +  t)H{K,  a  +  t)dt 
(78)  ^'(X„)  = -^ 


IV 

The  Summability  and  Convergence  of  Important  Special  Developments. 

Developments  in  Terms  of  Bessel  Functions,  Legendre 

Functions,  etc. 

1.   Certain  Important  Sine  Developments. 

04.  As  the  simplest  application  of  the  preceding  general  results  to  well- 
known  developments  in  mathematical  physics,  we  now  turn  to  the  development 

00 

(79)  H  qn  sin  \nX, 

n=l 

where 

(80)  g„  =  2  j^-^^:pY)  X  •'■(■'■)  ^'°  ^"'^  ^'^ 
and  where  the  quantities  X„  are  the  positive  roots  of  the  equation 

(81)  z  cos  2  +  p  sin  2;  =  0, 

y  being  a  (real)  constant  =1=  —  1.^^ 

In  this  case  U{z,  x)  =  sin  zx  so  that  the  differential  equation  (53)  becomes 

(82)  -^  +  zm{z,x)  =  Q. 

Thus,  we  have  /v(.r)  =  1,  F(.r)  =  1,  Fi{x)  =  0  and,  as  appears  from  (80),  a  =  0, 
6=1. 

]\Ioreover,  equation  (61)  becomes  satisfied  identically  for  all  values  of  z  if 
we  place  h'  =  00,  while  equation  (62)  becomes  (81)  if  we  place  h  =  —p. 

Considering  then  that,  in  the  notation  of  §  60,  we  here  have 

u{z)  =  s  sin  s  +  2^  sin  2 

and  noting  that  the  solution  sin  zx  of  (82)  is  one  to  which  the  general  results 
obtained  in  §§  61,  63  apply,  we  may  write  by  use  of  (64)  and  (81), 

/"   .  „       -         1  [ d  sin  zx  d  sin  zx        .         d^sinzx'] 

I     snr  zxdx  =  tz-     — _ . sin  zx     -  _ 

Jo  2z  L      ox  az  azdx     J^=i 

1  siri   z 

=  -^[z  cos^  z  —  sin  2  cos  z  -\-  z  sin^  z]  =  ~c^^  [z^  +  p{p  +  1)] 

1^  It  will  be  noted  that  this  form  of  development  is  the  one  required,  for  example,  in  the 
problem  of  the  cooling  of  a  sphere  in  air  at  temperature  zero.  Cf .  Byerly's  Fourier  series  (Boston, 
1895),  Chap.  IV,  §  67. 

134 


and  hen: 


or,  since 


we  may  write 


X 


Certain  Sine  Developments  135 


sin^  \nxdx  =    .^^  2"  [V  +  p(2^  +  1)] 

•     2  \  ^ 

Sin''  A„  = 


Xn'  +   P 


I         Dill      /\n*C  Cc^U/  r\  /N      0       I  9\  • 

Jo  2(X„'=  +  p') 

Thus  it  appears  in  the  first  place  that  the  coefficients  g„  as  calculated  by  (55) 

agree  with  the  given  values  (80). 

Now, 

^(2)  —  2>  sin  2 
u  (z)  =  —  2  sin  z  +  cos  z  -j-  p  cos  z  =  —  2  sin  2  +  (1  +  p) , 

^"(2)  =  —  sin  2  —  2  cos  2  —  sin  2  —  p  sin  2  =  —  [2  sin  2  +  ii{z)], 

and  hence 

(33)  j  w'(Xn)  =  -  ~  [X„2  +  P(P  +  1)], 

L  u"(Kn)  =  —  2  sin  X„, 
so  that 


(84)  I     sin^  Xnir  dx  =  - 

Jo 


X„2  +  p(p+l)' 


Let  us  now  avail  ourselves  of  formula  (77).^''  In  order  to  do  this  we  are 
first  to  determine  the  function  \{/{z)  according  to  the  condition 

A  possible  choice  of  1^(2)  is  xpiz)  =  2^  +  p(p  +  1)  since  from  (83)  we  have 
u['(Knl  =  2X„ 

Assuming  that  \l/(z)  has  been  chosen  in  this  manner,  we  now  have  to  deter- 
mine a  function  ^(2)  according  to  the  condition  (78)  which,  by  means  of  (84) 
becomes  in  the  present  instance 

sin  Xno:  I    sin  X,i(a  +  t)dt 

e'(K)  =  2[x.2  +  p(p  +  1)] -^^ 


'/'(Xn) 

2  sin  X„a  I    sin  X„(a  +  t)dt. 
Jo 


20  DiNi  has  shown  through  an  elaborate  investigation  that  this  formula  will  always  lead  to  de- 
cisive results  whenever  the  solution  H{z,  x)  has  the  special  form  II (zx);  that  is,  when  the  variables 
2  and  X  enter  only  through  their  product.  (Cf.  "Serie  di  Fourier,  etc.,"  §§  97-109.)  The  well- 
known  developments  in  terms  of  Bessel  functions  form  a  special  class  of  this  kind. 


136  SUMMABILITY   OF   FoUllIER   SERIES   .^J^D   ALLIED    DEVELOPMENTS 

Hence,  let  us  take  6{z)  such  that 

e'{z)  =  2  sin  za  j    sin  z{a  +  t)dt  =    I    [cos  iz  —  cos  (2a  +  t)]dt. 
Jo  Jo 

In  particular,  let  us  take 

e{z)  =   j     I    [cos  tz  -  COS  (2a  +  t)z]dzdt  =   I       — ~  ^      dt. 

Formula  (77)  thus  becomes 

r  If      z'  +  p(p  +  1)       r  [sin  tz      sin  (2a +0^1  ,, , 

Jo   ^^'^'  "'  ^^'^^  ==  27^  X  [.cos2+2'sin.]2  i    L  ~1  2a  +  ^      J  ^^^' 


(«^)       =2^r^^x 


c„  [2;  cos  2  +  2?  sin  z] 

1  +  a;i(2;)  sin  tz 


dz 


Q^  [cos  z  +  ^2(2)  sin  2]^     t 

_    1     r _d^   r  [1  +  coi(2)]  sin  (2a+02j 
27ri  Jo   2a  +  ^  Jc      [cos  2  +  0^2(2)  sin  2p         ' 


where  the  contour  C„  is  so  taken  as  to  inclose  the  roots  Xi,  X2,  •  •  • ,  Xn  and  only 
these  roots  of  the  equation  ^(2)  =  0  and  where,  in  the  last  two  integrals  we  have 
placed  for  simplicity 

y(p  +  1)  ,.       V 

^^1(2)  =  ^2 '        '^'2(2)  =  -. 

We  observe  at  this  point  that  in  applying  Theorems  I  and  II  of  §§  51,  52 
to  the  function  (p{n,  a,  t)  of  the  present  development,  the  values  of  a  and  t  with 
which  we  shall  bi  concerned  are  such  that 

r  0  <  a'  <  a  <  '/  <  1, 
(86)  \   -  a^t^l  -  a, 

[0  <  a'  <  a  ^  2a  +  t  ^  1  +  a  <  1  +  b'  <  2. 

Returning  then  to  (85),  let  us  take  as  the  contour  Cn  the  rectangle  formed  in 
the  2  plane  {z  =  x  -\-  iy)  by  the  lines  z  =  x  -\-  ij,  z  =  x  —  ij,  z  =  iy,  z  =  k  ■\-  iy\ 
j  being  any  positive  quantity  arbitrarily  large  and  k  being  any  positive  quantity 
lying  between  X„  and  X„+i.  Now,  the  function  appearing  in  the  integrand  of 
(85)  is  an  odd  function  of  2  which  remains  finite  in  the  neighborhood  of  the 
point  2=0  since  p  ^  —  1.  Whence,  the  portion  of  the  integral  in  question  due 
to  integration  over  the  2/-axis  is  equal  to  zero.  Upon  the  sides  which  are  parallel 
to  the  a:-axis  we  have  dz  =  dx.  Whence,  considering  first  the  side  upon  which 
z  =  X  -\-  ij  the  last  integral  of  (85)  extended  over  this  side  becomes 


J_   ndt  r 
27rWo  ^Wo 


{1  +  coi}  {sin  Ax  cosh  Aj-{-  i  cos  Ax  sinh  Aj] 


where   yl  =  2a  +  ^,    Di=  cos  x  cosh  j  —  i  sin  x  sinh  j  +  C02,    D2  =  sin  x  cosh  j 
+  i  cos  X  sinh  j. 


Certain  Sine  Developments  137 

Now,  the  functions  wi  =  o)i(z),  coo  =  ^2(2)  are  each  less  in  absolute  value 
than  a  constant  (independent  of  z)  provided  \z\>Q=  fixed  mimber  >  0. 
Thus,  we  have  but  to  make  use  of  the  well-known  properties  of  the  hyperbolic 
functions  to  see  that  if  we  place  j  =  +  co  the  expression  above  wall  approach 
uniformly  the  limit  zero  for  all  a  and  t  satisfying  relations  (86). 

Similarly,  we  reach  the  same  result  for  the  last  integral  of  (85)  when  extended 
over  the  side  upon  which  z  =  x  —  ij. 

Turning  now  to  the  first  integral  in  the  second  member  of  (85)  extended  over 
the  sides  upon  which  z  =  a;  ±  ij,  we  note  that 


and  hence, 


sin  tz      sin  tx       .     .       .  sinh  tj 

— —  =  — - —  cosh  tjzti  cos  tx  - — - — 

sin  fe  ,     .       . .  •  1       . 

—z —  =  X  COS  tix  cosh  tj  ±  ij  cos  tx  sinh  tij, 


where  ti  and  ^2  are  values  lying  between  0  and  t.  Moreover,  for  all  values  of  t 
under  consideration  in  (86)  we  have  |  ^  |  <  1  so  that  if  we  place  j  =  -\-  <x>  as 
before,  the  first  integral  in  the  second  member  of  (85),  like  its  last  integral,  will 
approach  uniformly  the  limit  zero  for  all  values  of  a  and  t  concerned  in  (86). 

We  turn  then  to  the  consideration  of  the  last  member  of  (85)  when  extended 
over  the  side  of  the  rectangle  C„  which  is  parallel  to  the  ?/-axis.  Here  we  have 
z  =  k  -{-  iy,  dz  =  idy  and,  having  taken  i  =  +  00 ,  we  see  from  what  has  just 
been  said  that  for  all  values  of  a  and  t  in  (86)  this  member  reduces  to 


27r  Jo        J- 00 


1  +  ^1  sin  iz 

dy 


[cos  s  +  a;2  sin  2]^     t 
(87) 


_Jl   r      ^^       r  U  +  ^i)  sin(2a+02 
27r  Jo  2a+  tj_^      [cos  z -\-  0^2  sin  zf        ^' 


in  which  it  is  understood  that  z  =  k  -\-  iy. 

Now,  it  sufiices  for  our  purpose  to  examine  the  behavior  of  (87)  as  k  =  00 
and  we  may  take  for  k  any  number  which,  at  least  for  all  values  of  ii  greater 
than  some  fixed  value,  increases  indefinitely  with  n  without  at  any  time  being  a 
root  of  the  equation  m{z)  =  0;  i.  e.,  of  the  equation 

E  =  cos  z  +  coo  sin  s  =  0. 

Thus,  we  may  take  k  =  mr,  in  which  case 

jB^  =  [cos  (mr  +  iy)  +  (^2  sin  {mr  -\-  iy)]-  =  coslr  y[l  +  10)0  tanli  y]- 
and  hence 


138  SUMM ABILITY   OF   FoURIER  SERIES   AND   AlLIED   DEVELOPMENTS 


(88) 


1  If  .  2      ^  tanh^  y  +  2iooi  tanh^  y 

E'=  ^^YTyV  "''"'''' ^^""^y  ~  '''  1  +  2io^,  tanh  y  -  <.,'  tanh^  y 


1 
cosh^  y 


tanh  w       5 
1  +  7— -^  +  r.(, 


where,  upon  recalHng  the  form  of  0^2(2),  we  have  7  =  —  2ip  and  therefore  inde- 
pendent of  z,  while  5  depends  upon  z  but  has  a  modulus  less  than  a  certain  fixed 
number  M  for  all  values  of    |  z  |  >  a  fixed  number  ko. 
Thus  expression  (87)  becomes 


(89) 
where 


y^'IK^+^^^^'^^+W 


27r 


sin  tz 


dy 


z^  J  t  cosh^  y 

-2^1   2^+lLV'+^'"^^^  +  ^0       cosh^^      ^^' 

1  +  -  tanh  ?/  +  -J  =  [1  +  coi]     1  +  -tanh  2/  +  -J  , 
2  z"  L         ^  ^"  J 

so  that  5i  like  5  has  a  modulus  less  than  some  constant  Mi  when  1 2 1  >  a  fixed 
number  =  ki. 

Considering  now  the  terms  in  (89)  which  have  2-  in  their  denominator,  we  see 
that  for  all  values  of  a  and  t  in  (86)  these  terms  approach  uniformly  the  limit  zero 
as  A;  =  CO .     Thus,  since 


(90) 


sin  (2a  +  0^ 


cosh^  y 


^  sm  {2(x  +  t)k  — 


cosh^  y 


+ 


sinh  (2q;  +  Qy 

cos  (2a  +  i)k r9 i 

cosh^  2/ 


where  0  <  a'  ^  2q;  +  i  ^  1  +  6'  <  2  we  have,  however  great  1 2 1  may  be, 

sin  {2a  -\-  t)z 


so  that 


cosh^  y 

1  r      dt       r*  5i  sin  (2a  +  t)z 
-Jo    2a+U-co22 


<2, 


27r  Jo    2a  +  ^  J_oo  2"        cosh^  ?/ 
In  like  manner,  noting  that 
sin  tz      sin  tk 


dy 


.   1     C^K..    r      dy 
sinh  ty 


Ml 

a'k' 


(91) 


cosh  <?/  +  i  cos  i^- 

Z  V  C 

=  ^*  cos  ^i/w  cosh  ty  -\-  iycos  tk  cosh  t2k, 
where  ti  and  ^2  lie  between  0  and  t,  and  recalling  that  for  all  values  of  t  under 


Certain  Sine  Developments  139 

consideration  we  have  |^|  <  1,  we  may  write 

II  r^.  r  ii     sinfe     .     ^Mi  f"       r      dy      ^  My 
27r  Jo  ^^  1.  z'  t  cosh^  2/  ^^  =   TT  X       J-»  /^-^  +  2/^        ^'   ■ 


Similarly  it  appears  that  the  derivative  with  respect  to  t  of  each  of  the  same 
terms  of  (89)  has  the  properties  just  indicated. 
Thus  (89)  reduces  to  the  form 


1    (••     dt       f/,    ,y(k-iy)^     ,     \  sin  (2a  +  t)k  cosh  (2a +  t)y  , 

(92)  -2ii  2^+'tL  V  +  l^T^*^"*^  V cosl?^ '^' 

i    r     dt       r(,   ,    y(k-iy)^     ,     \  cos  (2a  +  t)k  sinh  (2c,  +  t)y  , 

+  A  (a,  t,  k), 

where,  for  all  values  of  a  and  t  under  consideration,  A(a,  t,  k)  and  dA(a,  t,  k)ldt 
converge  uniformly  to  zero  when  ^=00,  Or,  expanding  and  dropping  integrals 
which  vanish  identically  since  they  are  relative  to  odd  functions  of  ij,  (92)  assumes 
the  form 

J^   r'sin  kt       n  cosh  ty       _  ji   r' sin  kt       f""  y  tanh  y  cosh  ty 
2tX       t     "^^l^cosh^i/^^      27rJo       t     ^^J_Ak'+y')cosh'y^ 

yi  r        7    7    r*  ^  tanh  y  sinh  ty  , 
+  2rX   ^°'^''*L(1-^  +  /)(cosh'y'^^ 

1     r'sin  (2a+<)i  J,  P^osh  (2a+0s'  . 

(93)  -  2. 1        2.  +  ^       '' L  -    cosh',        'y 

ji   f  sin  (2a;  +  t)k       r       y       tanh  y  cosh  {2a  +  t)y 
+  27rJo         2a +t       "^^  X^k^' +  f  cosh^  y  "^^ 

yi  r  cos  i2a -\- t)k  .    f"      k       tanh  y  sinh  (2a+%^     ,    ,,      ,7. 

27rJo         2a  H-^  J-oo/'^-H-r  cosh^  y 

We  proceed  to  consider  separately  the  six  integrals  here  appearing. 
The  first  may  be  put  into  the  form 

(-)  ^-^-fft7-  +  fJ''/.«-/'(0)l£?<^^ 

where 

,    ,       sin  t   r^cosh  ty  . 


140  SUMMABILITY    OF   FOUIRER   SERIES    AND    AlLIED    DEVELOPMENTS 

Whence,  if  ^  >  0  the  limit  of  the  first  term  in  the  last  member  as  ^  =  oo  is 
/i(0)/4  (of.  Appendix,  Lemma  II).     But 

and  hence  as  k  =  <x>  the  term  just  mentioned  approaches  the  limit  ^  when  ^  >  0. 
Again,  by  breaking  up  the  integration  in  the  last  term  of  (94)  into  that  from 
t  =  0  to  t  =  7]  plus  that  from  t  =  rj  to  t  =  t  (rj  arbitrarily  small  and  >  0)  and 
obser\ing  that  the  function /i(^)  has  limited  total  fluctuation  in  the  neighborhood 
of  the  point  t  =  0,  it  follows  that  the  same  term  approaches  the  limit  zero  as 
k  =  00  (see  Appendix,  Lemmas  I,  III). 

Likewise,  if  i  <  0  we  obtain  lim  Zi  =  —  ^. 

A-=oo 

The  second  and  third  integrals  of  (93)  may  be  reduced  respectively  to  the 
forms 

yi    r'  sin  kt  ,    T"      k^      y  tanh  y  cosh  ty 
~  2Tk  X    ~kr  "^^  J_„  FT?  ^osh^^^         '^^' 


r'        7    7    r"      ^'"      y  tanh  y  cosh  tiy 

I    cos  kt  at  I     7^—; — , , ^, ay, 

Jo  J-^k^+y  cosh^w  ^' 


where  tx  is  a  quantity  lying  between  0  and  t.     Since  we  have  alwaj^s 

P  ,  sin  kt 

^1.       -T:r-<h 


k^  +  y^^    '  kt 

it  thus  appears  that  the  limit  approached  by  each  of  these  integrals  as  A;  =  oo 
is  equal  to  zero. 

In  order  to  study  the  fourth  integral  of  (93)  let  us  make  therein  the  substi- 
tution 2a  -]-  t  =  2  —  r.     Since  k  =  7ir  the  integral  in  question  becomes 

27r  J2(i_,)  2  -  r      J_^        cosh^  y  ^' 

in  which  it  is  to  be  noted  that  for  all  values  of  a  and  t  in  (86)  the  quantity  r  is 
positive  {I  -  h'  <  T  <2  -  a'). 

The  expression  (95)  is  of  the  form 


(96) 
where 


If      .  .  s  sin  kr 

^TT  J 2(1. -a)  Sm  T 

We  have  now  but  to  apply  Lemma  I  of  the  Appendix  to  the  integral  (96) 
in  order  to  see  that  for  all  values  of  a  and  t  with  which  we  are  concerned  the 
expression  (96)  converges  uniformly  to  zero  when  k  =  oo . 


Certain  Sine  Developments  141 

Finally,  the  fifth  and  sixth  integrals  of  (93)  are  readily  seen  to  approach  the 
limit  zero  when  ^-  =  <»  and  the  convergence  is  uniform  for  all  values  of  a  and  t 
entering  into  (86),  since  we  have  always  \2a  -^  t\<  2. 

In  summary,  then,  the  present  function  (p{n,  a,  t)  satisfies  relation  (I)  of 
Theorem  I,  §  51,  it  being  understood  that  we  here  have  a  =  0,  h  =  1. 

We  turn  therefore  to  a  consideration  of  relation  (II)  of  the  same  Theorem. 
This  relation  is  at  once  seen  to  be  satisfied  since,  as  just  shown,  all  the  integrals 
of  (93)  converge  uniformly  to  zero  for  all  a  and  t  under  consideration  except  the 
first,  and  this  integral  satisfies  relation  (II)  of  the  theorem  by  virtue  of  Lemma  III 
of  the  Appendix. 

Again,  relation  (III)  of  Theorem  I,  §  51  is  readily  seen  to  be  satisfied  in 
the  present  instance  upon  noting  that  the  function  (p{n,  a,  t)  is  here  equal  to  the 
derivative  with  respect  to  t  of  the  expression  (93)  and  that  dA(_a,  t,  k)/dt  con- 
verges uniformly  to  zero,  as  already  pointed  out,  when  k  =  co . 

Before  summarizing  these  results  into  a  theorem  respecting  the  series  (79) 
we  turn  to  consider  the  application  which  may  be  made  in  the  present  instance 
of  the  general  Theorem  II  of  §  52,  thus  arri^dng  at  certain  results  concerning  the 
summability  of  the  same  series.  In  view  of  the  existence  already  demonstrated 
of  relations  (I)  and  (III)  of  §  51  it  will  here  suffice  to  consider  whether  relation 
(II)'  of  §  52  is  here  fulfilled.     Moreover,  the  properties  of  the  integral 


1     \^{n,  a,  t)  \dt; 

^0 


(98)  \^{n,a,t)\dt;         -e^t^e 

of  the  present  development  are  readily  obtained  from  the  expression  (93).  In 
fact,  in  order  to  be  assured  of  the  desired  properties  of  (98),  it  suffices  to  show 
that  each  of  the  seven  terms  of  (93)  when  affected  by  the  operation 

1         71 

-T, 

fl  71  =  0 

has  these  properties,  it  being  understood  that  absolute  values  are  employed  under 
each  integral  sign  and  in  the  integrals  which  constitute  the  expression  A(a,  t,  k). 
For  the  sake  of  simplicity  and  also  because  the  indicated  studies  are  readily 
carried  out,  though  the  forms  in  (93)  are  complicated,  we  shall  here  suppress  the 
details,  noting  simply  that  the  desired  result  follows  in  each  case  when  we  make 
use  of  Lemmas  IV  and  V  of  the  Appendix  and  make  use  also  of  (90)  and  (91) 
in  the  study  of 

-J2A{a,t,k). 

n  n=0 

We  turn  then  to  note  the  application  of  Theorem  \l  of  §  55  to  the  present 
development  in  order  to  ascertain  the  limit  approached  by  the  series  (79)  when 


142  SUMMABILITY   OF   FoURIER   SERIES   AND   ALLIED    DEVELOPMENTS 

For  this  purpose  we  first  observe  that  the  integral 

<p{n,  1,  t)dt 


is  here  obtained  by  placing  a  =  1  in  the  expression  (93).  In  the  resulting  new 
expression  the  first  three  integrals,  when  considered  for  values  of  t  such  that 
—  1  ^  ^  ^  —  e  are  readily  seen  to  have  the  properties  already  obtained  for  the 
corresponding  integrals  for  the  case  0  <  a  <  1  (in  which  case  —  \  <  t  instead  of 

The  fourth  term,  however,  does  not  approach  the  limit  zero  in  case  o:  =  1 
since  the  lower  limit  of  integration  in  (95)  is  now  equal  to  zero  so  that  the  reason- 
ing before  employed  can  not  be  used.  The  resulting  integral  now  assumes  the 
character  of  the  first  integral  of  (93)  and  if  treated  in  the  manner  naturally 
suggested  by  the  analysis  of  that  integral  we  find  directly  that  for  the  values  of  r 
under  consideration  the  limit  approached  as  Z:  =  oo  is  —  /i(0)/4  where  /i(0) 
is  to  be  determined  from  (97).  In  order  to  find  the  value  of /i(0)  it  is  desirable 
to  make  first  the  following  general  observation: 

If  (p{y)  is  a  function  of  the  real  variable  y  which,  together  with  its  first  deriva- 
tive, is  finite  for  all  values  of  y  then,  for  any  number  6  such  that  \Q\<  2  we  shall 

have 

f"     ,  ,  cosh  dy  ,  e        f"    ,,  ,  sinh  dy  , 

L  ^^^^  cosh^"^  ^^  =  4^=tJ_„  ^  (^^  cosh^y  ^y 
(99) 

2        r    ,,  ,  cosh  %  sinh  2/  ,     .       6        T"    ,     Posh  0?/  , 

+  ^ZTe^  J_  ^  (2/)        cosh^^       ^^  +  4^0^  L  ^^^^  cosF"^ ^y- 
In  fact,  integrating  once  by  parts  we  obtain 

n    ^  -cosh  02/  1    n         .sinhgy 

noo^  L^'^'^^^o^y'y^-'eL^^y^^^^y'y 

^^'"'  2  f    ,  ,  sinh  By  sinh  y  , 

and  in  like  manner  we  may  obtain  also 

f "     ,  ,  sinh  By  sinh  y  ^  1   f"    ,.  x  cosh  dy  sinh  y  , 

J_,  ^'^^   cosh'  y       '^y=--e}_J  (^>  ~^.\fy~-  "^y 
(101) 

,  2  f"  ,  ,  cosh  $y  ,        3  f"  ,,  ,  cosh  By  , 

^-eL-'^y^  J3ii?i  ^y-elj  ^y^  ^o^y  ^y- 

Whence,  upon  combining  (100)  and  (101)  we  obtain 
P    ^     nosh  02/  1    f"  sinh  02/  2    T*  cosh  0y  sinh  2/ 

4    f "    ,  ,  cosh  dy  ,         6    r=°    ,  ,  cosh  6y  . 
+  6^  L  ^^y^  c^sh^y  ^y  '  e-^  L  ^^^^  cosh^^  ^' 
and  this  equation  at  once  gives  (99). 


Certain  Sine  Developments  143 

Similarly,  we  may  find  an  analogous  form  for  the  integral 


f 

»/  — c 


^  sinh  dy  ^ 


Thus,  for  the  function /i(0)  where /i(t)  is  defined  by  (73)  we  may  write 
,^^^       ..      sinr    f°°cosh(2-T)y^       ..      f        ^  sin  r  f  °°cosh  (2-r)y  ^   ] 

_3  p  cosh  2y       _3/  r°°     dy  ptanh^y      \ 

~  4  J_„  cosh-*  y    ^  ~  4  \  J_„  cosh^  y      J_^  cosh^  y  ^  J 

=  I  ( [tanh  yr^  +  i  [tanh^  2/]^„  )  =  2. 

As  to  the  fifth  integral  of  (93)  when  a  =  1,  the  values  of  t  to  be  considered 
are  as  before  those  for  which  0  ^  t  ^  —  1  and  for  these  we  have 

|2a+  t\=  |2  +  ^!^2 

instead  of  [2  +  ^|  <  2.  The  reasoning  employed  in  studying  the  corresponding 
integral  when  0  <  a:  <  1  can  not  therefore  be  employed.  However,  if  we  break 
up  the  integral  in  question  into  that  from  t  =  0  to  t  =  ei  plus  that  from  t  =  ei 
to  t  =  t  (ei  arbitrarily  small  but  >  0)  the  last  of  the  two  integrals  thus  obtained 
will  have  the  limit  zero  as  A:  =  oo  since  for  the  values  of  t  concerned  we  have 
|2q:  +  ^|^2—  ei<2;  while  the  first  of  the  same  integrals  may  be  made  arbi- 
trarily small  with  ei  since  by  placing 

tanh  y  cosh  (2  +  t)y  cosh  {2  +  t)y 

^^^y =  *'<2') -^i^lT     •         ^(^/^  =  *•"'•>  2' 

and  applying  (99)  we  see  that  the  integral 


L 


y  cosh  (2  +  t)y        , 


-00  ^'^  +  2/^  cosh^  y 

remains  less  in  absolute  value  than  a  constant  independent  of  ei  for  all  values  of  t 
such  that  —  €i  ^  ^  ^  0. 

Similarly,  when  a  =  1  the  sixth  term  of  (93)  may  be  neglected  in  the  limit 
as  A:  =  CO . 

Thus  condition  (1)6  of  Theorem  VI,  §  55,  becomes  satisfied  in  which  in  the 
present  instance  we  have  6^2=  —  ^  ~  2  —  —  1(q:=  1,  a=0,  6=  1). 

Relations  (II);,  and  (111)6  of  §  55  as  well  as  (11)6'  are  now  readily  seen  to  be 
satisfied  (as  in  the  studies  already  carried  out  in  connection  with  (93))  so  that 
by  virtue  of  the  general  theorems  of  §§  51-55  we  reach  in  summary  the  following 


144  SUMMABILITY    OF   FoURIER   SeRIES    AND    AlLIED    DEVELOPMENTS 

Theorem.     Iff(x)  remains  finite  throughout  the  interval  (0,  1)  with  the  possible 
exception  of  a  finite  number  of  points  and  is  such  that  the  integral 

(102)  r\f{x)\clx 
exists,  then  the  series 

(103)  2  qn  sin  \nX 
in  tvhich 


On  =  2  -v  o   i" — 7 — ; — Tn   I    f{x)  sin  \nxdx:        p  =  constant  =}=  —  1, 
Xn  +  p(p  +  1)   ' 


I    f{x)  sin  \nxdx; 
Jo 


\n  being  the  nth  positive  root  of  the  equation 

z  cos  z  -\-  p  sin  2  =  0, 

loill  converge  at  any  point  x  (0  <  .r  <  1)  in  the  arbitrarily  small  neighborhood  of 
which  f(x)  has  limited  total  fluctuation,  and  the  sum  will  be 

H/(-^-0)+/(.r+0)]. 

Moreover,  the  convergence  loill  be  uniform  to  the  limit  f{x)  throughout  any  in- 
terval (a',  b')  enclosed  U'ithin  a  second  interval  (ffi,  bi)  such  that  0  <  ai  <  a'  <.  b' 
<  bi  <  1  provided  that  f(x)  is  continuous  throughout  (a',  b')  inclusive  of  the  end 
points  X  =  a',  X  =  b'  and  has  limited  total  fluctuation  throughout  (ai,  bi). 

Also,  if  f{x)  remains  flnite  throughout  the  interval  (0,  1)  with  the  possible  ex- 
ception of  a  finite  number  of  points  and  is  such  that  the  integral  (102)  exists,  then 
the  series  (103)  ^vill  be  summable  (r  =  1)  at  any  point  a:  (0  <  a:  <  1)  at  which  the 
limits  f{x  —  0),  f{x  +  0)  exist  and  the  sum  will  be 

i  [/(.r-0) +/(.!• +  0)]. 

Moreover,  the  summability  ivill  be  uniform  (§  45)  to  the  limit  f{x)  throughout 
any  interval  {a',  b')  such  that  0  <  a'  <  b'  <  1  provided  that  at  all  points  ivithin 
{a',  b'),  inclusive  of  the  end  points  x  =  a',  x  =  b',  the  function  f(x)  is  continuous. 

Under  the  same  conditions  for  f{x)  when  considered  throughout  the  whole  interval 
(0,  1),  the  series  (103),  when  considered  for  the  value  x  =  \,  will  converge  to  the  limit 
/(I  —  0)  provided  f{x)  is  of  limited  total  fluctuation  in  the  neighborhood  at  the  left 
of  the  point  x  =  \  and  will  be  summahle  (r  =  1)  to  the  limit  f  {I  —  0)  ichenever  this 
limit  exists. 

65.  It  may  be  observed  that  in  the  exduded  case  for  which  p  =  —  1  the 
methods  which  we  have  followed  may  be  readily  altered  so  as  to  yield  corre- 
sponding results.  In  this  case  the  integrand  of  (85)  has  a  pole  at  the  point 
2  =  0  so  that  this  point  should  be  excluded  from  the  contour  Cn-  Supposing 
this  to  have  been  accomplished  by  means  of  a  small  semicircle  extending  to  the 
right  of  z  =  0,  we  may  then  take  as  Cn  the  resulting  contour  in  part  rectangular 


Developments  in  Bessel  Functions  145 

and  in  part  semicircular.  If  the  integrations  be  now  carried  out  as  before  over 
the  respective  portions  of  Cn,  that  arising  from  the  semicircle  will  be  equal  to 
—  ^r  where  r  represents  the  residue  of  the  integrand  of  (85)  corresponding  to 
the  pole  2=0.  Except  for  this  auxiliary  term,  the  reductions  are  the  same  as 
before,  so  that  in  applying  the  general  theorems  of  §§  51-55  we  encounter  an 
application  of  the  remark  (3)  in  §  56.  A  similar  instance  will  occur  in  connec- 
tion with  the  developments  in  terms  of  Bessel  functions,  to  which  we  now  turn, 
and  in  that  case  we  shall  elaborate  the  consequences  at  some  length,  though 
such  studies  will  be  omitted  for  the  sake  of  brevity  in  connection  with  the  present 
series  (103). 

2.   The  Developments  in  Terms  of  Bessel  Functions. 

66.  As  a  second  application  of  the  general  results  obtained  in  §§  51-55  we 
shall  now  consider  certain  developments  in  terms  of  the  function  P^(z)  defined 
by  the  equation 

where  Jy{z)  represents  Bessel's  function  of  order  v.  The  developments  in 
question  are  closely  related  to  the  well  known  developments  for  an  arbitrary 
function  in  terms  of  Bessel  functions  and  at  once  yield,  as  we  shall  show,  results 
of  considerable  generality  concerning  the  summability  and  convergence  of  the 
latter. 

For  the  function  P^(2;)  as  thus  defined,  we  have,  when  v  4=  negative  integer. 


P  (^^)  =  '^^^a^  = ^ r  1 v"">      j_ 

"^^^        (zxY       2T(z/+l)L         '"''-   '    "" 
(104) 


{zxY  ~2T(z.+  l)L         22(z.+  l)"^24-2!(^+l)(^+2) 

{zxf  _ 


26.3!(^+l)(^  +  2)(^  +  3) 
while  the  equation  (53)  becomes 


^  -^+22a;2''+ip^(s.r)  =  0 


dx 
or,  placing  for  brevity  P^(2;.r)  =  P, 

d-P  dP 

(105)  ^^  +  (2,  +  1)  —  +  22,.p  ^  0. 

Taking  a  =  0,  b  =  1,  the  development  (54)  in  terms  of  the  functions  P,,(X„.r) 
becomes 

(106)  llqnP.iKx), 

11 


146  SUMMABILITY   OF   FOURIEK  SERIES   AND   AlLIED   DEVELOPMENTS 

where 


(107) 


?n  = 


(KnX)dx 


Equations  (60)  become 


Jo 


x-''+'PJ'{\nX)dx 


(108) 


,^.2.+i      p^(X„.r)  -  hT^iKx)  =  0    ichen    x  =  0, 
ox 

3,2.+i      p^(X^3.)  _  /iP^(x,,r)  =  0     when    x  =  1. 
ox 


Of  these  the  first  is  seen  to  be  satisfied  identically  for  all  values  of  z  if  we  place 
/?'  =  0  and  assume  j'  >  —  1,  while  the  second  gives  as  the  equation  u{z)  =  0 
(cf .  §  64)  of  the  present  developments, 

dPM 


u{z)  =  z 


dz 


-  liPXz)  =  0. 


We  may  therefore  apply  (64)  and  write 

dP    dP 


jr'.-.p.w..  =  ^['/-'^-p^l 


or  smce 


we  have 


dP      zdP 


dx      dz 
a^P       z  d^P 


d-P 
dzdx  Jx=i 


dx       X  dz  '        dzdx      x  dz^ ' 

dPY_l^dP      z  ^d^P 


(109) 


L 


by  (105). 

Thus,  \i  h  =  ±  CO  so  that  u(z)  becomes  simply  P^iz),  we  have 

(110)  f  x-^'+'PJ'iKx)dx  =  i  (£  ^.(2)  )]^^  =  h^'O^nY 

and,  since  we  then  have  by  (105), 

it  appears  that  if  we  wish  to  apply  (77)  in  the  study  of  the  function  (p(n,  a,  t) 
of  the  present  developments  we  may  take  at  once  \l/(z)  =  l/s-""*"^  and  ^(2)  such  that 

(111)  d'{z)  =  2z'"'+'P{az)   r  («  +  ty-''+'P{(a  +  t)z}dt;         P  =  P,. 

Jo 


Developments  m  Bessel  Functions  147 

On  the  other  hand,  if  h  be  finite  so  that  u{z)  =  zP'{z)  —  hP(z),  we  shall 
have  by  (109) 

(112)  J\'-'+'F-(Knx)dx  =  ^^^  {h(2v  +  h)  +  X„2}. 

Now,  we  have  by  (105) 

u'{z)  =  zP"iz)  +  (1  -  h)P'iz)  =  -  {2p+  h)P'(z)  -  zP{z), 

i2v-{-  1)(2j/+  h) 
u"{z)  =  -  (2^  +  h)P"{z)  -  zP'iz)  -  P(z)  =  ^      ^    ^^      ^    ^  P'iz) 


Whence, 


-  zP'{z)  +  {2v  +  A  -  l)P(s). 


so  that 


An 

^"W  =  ^^¥  {^(2^  +  1)(2^  +  /^)  +  (2^  -  1)X„2}, 

u'{Kt ^  2{/i(2^  +  /i)  +  Xn'l 


r  a:2''+ip2(X„.T)(/.r 
w''(Xn)  1  Klv  +  l)(2j'  +  /O  +  (2^/  -  l)Xn2  2j/  +  1   ,  2X„2 


W'(X„)  Xn  /i(2j'  +   /l)   +  Xn'  Xn  '     ^(2j/  +   A)  +  Xn^  * 

Thus,  in  order  to  satisfy  the  conditions  relative  to  -^{z)  in  the  present  case 
we  should  take  it  so  that 

lA'(z)  2j.  +  1  2z2 


i^iz)  Z  '     ]l{2v^  /0  +  22- 

Let  us  therefore  take 

]i{2v  +  A)  +  z2 


i^iz) 


z 


,2»'+l 


in  which  case  it  appears  that  we  may  take  d(z)  as  before,  viz.,  such  that  equation 
(111)  is  satisfied. 

Now  we  have  from  (105) 

with  a  similar  equation  obtained  by  replacing  a  by  o:  +  i^-     Hence, 

[{a  +  ty  -  a'']z'''+'P(az)P{{a  +  t)z] 


148  SUMMABILITY   OF   FoUKIEK   SeRIES   AND   AlLIED    DEVELOPMENTS 

Placing  for  convenience  a  -\-  t  =  (3  and  letting  accents  represent  differentiation 
with  respect  to  z,  we  may  therefore  write 

r z'''+'Piaz)P{^z)dz  =  J^,[P'(az)P(l3z)  -  P'{^z)P(az)] 
Jo  pa. 

so  that  both  when  h  is  finite  and  when  infinite  we  may  take 

(113)  e{z)  =  2z^"+'  X'^^^-  [P'(az)P{(3z)  -  P'mP(az)]dt. 

Upon  noting  the  analytic  properties  of  the  functions  \p(z)  corresponding  to 
the  two  above  mentioned  cases  and  of  the  function  d(z),  it  appears,  upon  applying 
(77)  that  the  integral  (36)  of  the  present  developments  will  be  given  by  the  ex- 
pression 

H    /,2.+i  r  P'(az)Pm  -  P'mPiaz) 


1    n    8^"+^         r 

(114)  -:  ^2 Jt 


P\z) 


dz 


or 


according  as  ii{z)  =  P(z)  or  2i{z)  =  zP'{z)  —  hP(z). 

It  is  to  be  noted  also  that  in  the  developments  (106)  we  shall  have  by  (110) 
and  (112) 

2       r^ 

Qn  =  p//^   s2    I     f{x)x^''+^P(KnX)dx 

or 

2x  ^  r^ 

^"  "  [h{2p  +h)  +  \J]P'{K)  Jo  /(^)^"'"^'^(^-^)^^ 

according  as  the  quantities  Xi,  X2,  •  •  •  are  the  roots  of  P{z)  =  0  or 

zP'(z)  -  hP{z)  =  0. 

These  results  premised,  let  us  now  consider  the  rectangle  in  the  z  plane  whose 
vertices  are  the  points  z  =  ij,  z  =  k  -\-  ij,  z  =  k  —  ij,  z  =  —  ij,  j  being  any 
positive  quantity  arbitrarily  large  and  k  being  any  positive  quantity  lying  be- 
tween Xn  and  Xn+i  where  Xi,  X2,  •  •  •  represent  the  successive  positive  roots  of  the 
equation  P{z)  =  0  or  zP'{z)  —  hP(z)  =  0  according  as  we  are  dealing  with  (114) 
or  (115).  From  the  boundary  of  this  rectangle  let  us  exclude  the  point  2=0 
by  means  of  a  small  semicircle  of  radius  rj  and  let  us  now  take  the  resulting 
contour  in  part  rectangular  and  in  part  semicircular  as  the  contour  of  integration 

Now,  the  function  appearing  in  the  integrand  of  either  (114)  or  (115)  is  an 
odd  function  of  z  and  hence  the  two  portions  of  these  integrals  extended  along 
the  y  axis  mutually  destroy  each  other,  while  in  either  case  the  portion  extended 


Developments  in  Bessel  Functions  149 

along  the  semicircle  may  be  made  arbitrarily  small  with  rj  unless  in  (115)  we 
have  ^  =  0.  In  this  exceptional  case  the  integrand  of  (115)  has  a  pole  of  the 
first  order  at  the  point  2=0  and  hence,  upon  applying  Cauchy's  integral  the- 
orem, the  value  of  the  contribution  to  (115)  arising  from  the  semicircle  in  question 
becomes 


-  {2p  +  2)    r  ^^''+^dt  =  q;2''+2  -  /32''+2. 
Jo 


(117) 


In  order  to  discuss  the  remaining  portions  of  the  integrals  (114)  and  (115) 
we  shall  now  make  use  of  the  following  established  result  :^^ 

"  Representing  by  J^(z)  Bessel's  function  of  order  v  we  shall  have  when 
V  >  —  ^  and  z  has  any  value  except  zero  whose  real  part  is  positive  or  zero 

(116)  JM    =    -  -i-[^i(z)e^(-«2.-l)/4].)  _   ^^(^Sjg-i(.-[(2.-l)/4].)-|^ 

'\2Trz 
where 

and  where,  when  lz|  is  sufficiently  large,  these  expressions  *Si  and  *S2  may  be 
expanded  into  the  forms 

(118) 

^f^      "^^        r(^  +  i  +  n)  1 

^'^'^  =  h  iXM^I)i>TI^^)  (2^-  +  ^'^''  '^' 

in  which  m  is  any  positive  integer  and  in  which  the  expressions  9i(;:,  v)  and 
62(3,  v)  become  infinitesimals  of  order  as  high  as  the  7?zth  when  |s|  =  00  and,  at 
least  when  v  >  -\-  ^,  possess  first  derivatives  which  as  |  z  |  =  00  become  infinitesi- 
mals of  order  as  high  as  the  {m  +  l)st." 
Placing  —  i  =  e'"'''^  in  (116)  we  obtain 

JXz)  =  -7^[5i(z)e*^^-f^'''-^^/''^'^)+  52(s)e-«^-«'^-^^/'i'^i 

Whence,  upon  expanding  and  making  use  of  (104),  we  have 

^^(^)  =  2^)>^  [  {^^(^)  +  ^2(^)}  cos  (2  -  ^tt) 

+  i{Sr(z)  -  S,{z)]  sin  i^z  -  — ^tt)], 

21  Cf.  H.  Weber,  Math.  Annalen,  Vol.  37  (1890),  pp.  404-416.  The  facts  which  we  shall 
state  regarding  the  derivatives  of  61(2,  p)  and  02(z,  p)  are  not  explicitly  obtained  by  Weber,  but 
follow  at  once  from  his  analysis. 


150  SUMMABILITY   OF   FOUEIER   SeRIES   AND   ALLIED   DEVELOPMENTS 

SO  that  by  (118)  we  may  write  when  v  >  ^  and  when  the  real  part  of  z  is  positive 
(or  zero) 


(120) 


^.(2)  =  '\^^^^^H^)L^^'^  '^^'''^^  cos\^z-~'^j—Trj 


.     /         2p-\-l     \ 

+  ^{Z,  v)  sin  I    Z ^ TT    I 


where  the  functions  e{z,  v)  and  f(z,  v)  become  infinitesimals  of  at  least  the 
second  and  first  orders  respectively  as  1 2 1  =  co  and  possess  first  derivatives  which 
as  I  z  I  =  00  become  infinitesimals  of  at  least  the  third  and  second  orders  respec- 
tively. 

Moreover,  by  use  of  the  relation  Py{z)  =  {2v  +  2)P^i{z)  —  z^P^^iz),  we 
may  readily  show  that  (120)  holds  true  for  all  values  of  v  for  which  P^(z)  has  a 
meaning — i.  e.,  unless  1/  is  a  negative  integer. 

Furthermore,  since  P'{z)  =  —  zP^^i{z)  we  see  that  unless  j^  is  a  negative 
integer  we  may  write 


(121) 


p/(2)  =  _  y^l^-—^^Y  [l  +  y]{z,  v)]  sin  [z  -  -^^-^-^J 


(         2?/+  1    \1 

+  Q{Z,   V)  cos  I    Z ^ TT  J  J 


where  77(2,  v)  and  ^(2,  v)  have  the  properties  mentioned  above  for  e(2,  v)  and 
f  (z,  v)  respectively. 

Equations  (120)  and  (121)  having  been  obtained,  we  return  now  to  the  dis- 
cussion of  (114)  and  (115)  when  the  indicated  integration  is  extended  over 
the  portions  of  C„  remaining  after  removing  the  semicircle  of  radius  77  and  the 
portions  of  the  y  axis.     Placing  for  brevity 

21/ +1  /2 

a=--^-7r,         c=^-, 

we  have  by  (120)  and  (121)  for  all  values  of  z  upon  these  portions  of  C„,  unless 
a  =  0  or  i8  =  0, 

(122)  P{pd)  =  ,     w+(i/2)[{l  +  €(az)}  cos  {az  -  a)  +  ^{az)  sin  (az  -  a)], 

\CX.Z) 

(123)  P'{az)  =    ,    .,!■(! ;2)  [  U  +  r){az)  \  sin  {az  -  a)  +  e{(xz)  cos  (az  -  a)], 

(124)  Pm  =        !^o/2)[{l  +  f(/32)}  cos  (iSz  -  a)  +  r(i32)  sin  {^z  -  a)], 

(125)  P'ifiz)  =    ,  ~,!a.) [  { 1  +  77(i3z) }  sin  (^z  -  a)  +  Sm  cos  (/32  -  a)], 

z{pz) 


Developments  in  Bessel  Functions  151 

and,  excluding  the  case  in  which  a  =  0,  we  observe  that  in  applying  Theorem  I 
of  §  51  to  the  integrals  (114)  and  (115)  in  question  the  values  of  a,  t  and  fi  =  a  ■}-  t 
with  which  we  shall  be  concerned  are  such  that 

0  <  a'  <  a  <  6'  <  1, 

—  a  =  t  ^  1  —  a, 

0<a'<a^a  +  p^l  +  a<l  +  h'<2, 

while  in  applying  Theorems  VI  and  VIII  of  §  55  for  the  case  in  which  a  =  1, 
we  shall  have  —  l^t^0,l^a-\-^^2. 

However,  when  ^  =  —  a  we  have  /3  =  a  +  i  =  0  so  that  expressions  (114) 
and  (115)  cannot  be  used  for  all  the  values  of  (3  with  which  we  shall  be  concerned. 
Let  us  therefore  exclude  for  the  present  the  value  t  =  —  a  from  our  investigations, 
treating  it  later  as  one  of  the  exceptional  values  of  the  t^^pe  mentioned  in  remark 
(1)  of  §  56.  Thus,  representing  by  ^  an  arbitrarily  small  positive  quantity,  we 
proceed  to  study  the  integrals  (114)  and  (115)  for  all  values  of  a,  t  and  jS  satis- 
fying the  relations 

0  <  a'  <  a  <  b'  <  1, 

(126)  -a+^^t^l-a, 

0<a'<Q;<a+^^a  +  ^^l  +  a<l  +  6'<2, 
or 

a  =  1, 

(127)  -  1  +  ^  ^  i  ^  0, 

1  ^  a  +  /3  ^  2. 

From  (122),  (123),  (124)  and  (125)  we  find  upon  performing  the  indicated  multi- 
plications that 

^^[P'(«.)P(^.)  -  P'mPiaz)]  =-,2.-^(^2  _^-^(^) 

X  [{a[l  +  €(l3zm  +  vic^z)]  -  (3d{(3z)Uaz)}  sin  (az  -  a)  cos  (/3s  -  a) 

-  1/3[1  +  e{az)][l  +  -ni^z)]  -  adiaz)^(^z)}  sin  (/Ss  -  a)  cos  {az  -  a) 

+  {a[l  +  7/(ce2)]r(/32)  -  ^[1  +  vm]tiaz)]  sin  (az  -  a)  sin  {(3z  -  a) 

+  {a[l  +  e(l3z)]d(az)  -  /3[1  +  e(az)]dm\  cos  (az  -  a)  cos  {^z  -  a)] 

—  iS. 


_/^Y+(i/2) 

o?)\a)  ^" 


1  sin  {az  —  a)  cos  (/Sz  —  a) 


22^+1(^2  _ 

+  B  sin  ((Sz  —  a)  cos  {az  —  a)  +  C  sin  {az  —  a)  sin  {^z  —  a) 
+  D  cos  {az  —  a)  cos  {^z  —  a)] 


152  SUMMABILITY   OF   FoURIER  SeRIES   AND   AlLIED    DEVELOPMENTS 

_  (3.  //Q\^+Ci/2) 


„  2H-U/Q2       n  ,       ,  [{A  -  B)  sin  (a  -  ^)z  +  {A -^  B)  sin  (a  +  /3  -  2a)z 

+  (C  +  D)  cos  (a  -  (3)2  +  (Z)  -  C)  cos  (a  +  i3  -  2a)2], 

where  ^,  5,  C  and  D  are  used  for  brevity  to  denote  the  respective  coefficients 
given  above  of  sin  {az  —  a)  cos  (/3s  —  a),  etc. 
Now,  we  may  write 

A-  B=  {cx  +  /3)[1  +  p,iz)],         A  +  B=  {a-  /3)[1  +  2^2(2)], 

C  +  Z)  =  (a  -  i3)p3(2),         C  -  2)  =  («  -  ^)p4(2), 

where,  recalUng  the  properties  of  the  functions  e{az),  ei^z),  etc.,  we  see  that  for 
all  values  of  a,  t  and  /S  in  (126)  and  (127)  and  for  all  values  of  z  now  under  con- 
sideration the  functions  pi(z),  ^2(2),  2^3(2)  and  2^4(2)  are  finite  and  vanish  uni- 
formly like  I/2-,  1/2^  1/s  and  I/2  respectively  as  |2|=  co.  We  note  also  that 
these  functions  may  if  desired  be  put  in  the  forms 

,  .       60  /  X       ^0  ,  .       do      eo  ^  s      fo  .   90 

3  2  <6  »i  til  Aj 

in  which  rfo  and  /o  depend  only  upon  a  and  jS  and  are  finite  for  all  values  of  a 
and  /3  in  (126)  and  (127)  while  &o,  co,  e^  and  <7o  depend  also  upon  2  but  for  all 
values  of  a  and  jS  under  consideration  and  for  all  values  of  s  under  consideration 
and  such  that  \z\>  ^  =  constant,  are  less  in  absolute  value  than  certain  con- 
stants independent  of  a,  (3  and  s.     For  the  same  values  of  2  we  have  also 

P(z)  =  -i;:f(i]^)  [1  +  e(2)][cos  (2  -  a)  +  u{z)  sin  (2  -  a)] 

zP'{z)  -  hP{z)  =  ^-^^)    1  +  77(2)  + 1  r(2)  I  [sin  (2  -  a)  +  ZJ(2)  cos  (2  -  a)], 
where 

,(.  0(2)4-^1  +  6(2)] 

f .       r(2)        -f  s    2 

^  ^^  i  +  7?(2)  +  ^r(2) 

Whence,  upon  recalling  that  a  —  /5  =  —  i,  we  see  that  whether  we  are  dealing 
with  (114)  or  (115),  the  portions  of  the  integral  arising  from  the  part  CJ  of  C„ 
now  under  discussion  will  be  of  the  form 

1_  p/^Y+g/^)     dt       r    sin  [(a  +  /3)2  -  2a] 


dz 


Developments  in  Bessel  Functions  153 

."+(1/2)     dt       r   (?2(z)  sin  [(a  +  ^)z  -  2a] 


1     f7j8\''+(i/2)     ^^       /> 


E' 


^2 


'^  2Tri 


TTlJo    \«/  a  +  jSJc^/  JS2 

l_   /-Y/3y+a/2)     ^^        /^    g4(2)  cos  [(g  +  ^)2  -  2a] 
+  27rJo    UJ  a  +  ^X/  ^^ 

where  the  functions  ^1(2;),  ^2(2),  93(2)  and  q^iz)  (Hke  the  functions  ^1(2),  2^2(2), 
etc.)  may  be  put  into  the  forms  61/2^  01/2^,  di/z  +  61/2^  and  /1/2  +  gi/z^  respec- 
tively and  where 

E  =  cos  (2  —  a)  +  co(2)  sin  (z  —  a), 
(129) 

E  =  sin  (z  —  a)  +  £0(2)  cos  (2  —  a), 

according  as  we  are  deahng  with  (114)  or  (115). 

Considering  first  the  portion  of  Cn  consisting  of  one  of  the  Hnes  parallel  to 
the  X  axis,  we  readily  obtain  as  in  §  64  the  fact  that  for  all  values  of  a,  /3  and  t 
in  (126)  each  of  the  integrals  in  (128)  when  extended  over  the  line  in  question 
approaches  uniformly  the  limit  zero  as  j  =  00 .  Thus  we  have  merely  to  con- 
sider (128)  in  which  z  =  k  -\-  iy  and  CJ  is  understood  to  extend  from  y  =  —  <x> 
to  2/  =  +  °°  along  the  line  2  =  ^  +  iy. 

Now,  from  the  manner  in  which  k  is  to  be  chosen,  we  see  from  (129)  that  we 
may  take  ^  =  nr  +  a  or  ^'  =  mr/2  +  a ;  (n  =  positive  integer)  according  as  we 
are  dealing  with  (114)  or  (115),     In  either  case,  equations  (129)  are  such  that 

1  7,5 

-^=  1  +  -tanh  y  +  -, 

t,"  Z  Z" 

where  y  is  independent  of  z  while  5  depends  upon  z  but  has  a  modulus  which  for 
all  values  of  2  under  consideration  is  less  than  a  certain  quantity  M. 
Thus,  (128)  may  be  written  in  the  form 

({^+|)cos[(a  +  /3).-2a]|^, 


+ 


+ 
where  62  has  the  properties  mentioned  above  of  61 


154        SuMMABiLiTY  OF  Fouhier  Series  and  Allied  Developments 

Considering  first  the  terms  of  (130)  which  have  z-  in  their  denominator,  we 
have  but  to  refer  to  the  discussion  of  similar  terms  in  (89)  in  order  to  see  that 
for  all  values  of  a,  /3  and  t  in  (126)  these  terms  have  uniformly  the  limit  zero  when 
^-  =  CO .     The  same  is  true  also  of  the  term 


X\a)  a  +  I3j_^z  cosh2  y   ^' 


since  we  have  cos  tz  =  cos  tk  cosh  ty  —  i  sin  tk  sinh  ty  and  we  know  that  when 
|/|  <  1  (as  is  the  case  in  (126))  the  integrals 


r"  coshj^  r°°  I  sinh  ty\ 

J_«,  cosh^  y   ^'         Xoo    cosh2  y     ^ 


have  a  meaning. 

Thus,  (130)  reduces  to 


.+(1/2)       /-»  /         y  \    sin  fe 


tm  '£('*>"") 


dy 


27r  Jo    \ « /  J-oo  \  ^  /  i  cosh-  y 

where  e/  depends  only  upon  a,  jS  and  t  and  is  finite  for  all  values  of  these  quan- 
tities in  (126)  and  where  A(a,  t,  k)  and  also  dA(a,  t,  k)/dt  depends  upon  a,  /3, 
t  and  z  but  when  considered  for  all  values  of  a,  /3  and  t  in  (126)  may  be  made 
(uniformly)  as  small  as  we  please  in  absolute  value  by  taking  k  sufficiently  large. 
Upon  placing  z  =  k  -{-  iy  and  recalling  the  values  which  k  may  assume; 
also  placing  for  convenience  a  +  /3  =  2  —  r  and  dropping  those  integrals  which 
vanish  identically  since  they  are  relative  to  odd  functions  of  y,  we  thus  obtain 
(131)  in  the  form 

1     rf^y-^^"^hmkt^    r  cosh  ty, 
2^1   [a)  ~r"^U-«c^sh^^^ 

_yi   r'  f^Y^^"^^^lM  n  y  tstnh  y  cosh  ty 
~  27r  Jo   U  J  t      J_„  (F  +  f)  cosh2  y  "^^ 

yi   r'  (^Y^^m  pfctanhysinh/y^ 

+  2^Jo    Uj         ^«^"J_„(F+F)icosh^/^ 

i-   r     /^V^^"'^sin^        r°°cosh(Q;  +  /3)y 
^■27r4_,U/  a  +  rJ-.        cosh'y        "^^ 


(132) 


r     /^V^^"'^sin_fcr        r°"       y       tanh  y  cosh  [a  +  f>)y 
L-.m)         a^rL^k'^f  cosh^?/  ^^ 


Developments  in  Bessel  Functions  155 

i_   r      /A*""^''^    /Cos^j     r°°fccosh  (a  +  i5)y 

27rJo(,_„)\Q;y  q:  +  |8       J_«,  (^- +  ^z^)  cosh^  ?/ 

the  upper  or  lower  sign  being  taken  according  as  we  are  dealing  with  (114)  or  (115). 

The  expression  (132)  may,  moreover,  be  used  to  determine  the  value  of  the 
integrals  (114)  and  (115)  corresponding  to  the  case  a  =  I.  In  fact,  when  a  =  1 
and  (3  and  t  are  confined  by  (127)  we  readily  see  that  each  term  of  (132)  con- 
tinues to  have  a  meaning. 

From  the  properties  already  found  of  the  integrals  in  (93)  it  now  appears  that 
the  second,  third,  fifth,  sixth,  seventh  and  eighth  integrals  of  (132)  when  con- 
sidered for  all  values  of  a,  ^  and  t  in  (126)  or  (127)  have  uniformly  the  limit 
zero  as  ^*  =  °o ,  while  if  we  treat  the  first  integral  as  w^e  treated  the  first  integral 
of  (93),  remembering  here  that  lim  (^(aY^^'^'^-^  =  1,  we  find  that  when  k=  ^  this 

integral  behaves  precisely  as  the  indicated  integral  of  (93) — i.  e.,  approaches  the 
limit  i  or  —  I  according  as  ^  >  0  or  i  <  —  0. 
Similarly,  the  integral 

J_   r      /^y+a/2)sin^T        r°°cosh  (a  +  /3)y 
27ria-„)U/  a  +  ^      .L        cosh^  2/  ^' 

like  the  fourth  integral  of  (93),  has  uniformly  the  limit  zero  if  a'  <  a  <  h'  while 
if  a;  =  1  it  has  the  limit  ^. 

Whence,  if  we  are  dealing  with  values  of  a,  (3  and  t  satisfying  (126),  the  ex- 
pression (132)  converges  uniformly  to  the  limit  |  or  —  |  when  k  =  <=o  according 
as  ^  >  0  or  ^  <  —  0,  while  if  a  =  1  and  /3  and  t  have  values  consistent  with 
(127)  the  same  expression  has  the  limit  0  or  1  according  as  we  are  dealing  with 
(114)  or  (115). 

Thus,  exception  being  made  of  the  case  h  =  0  in  the  integral  (115),  the  inte- 
grals (114)  and  (115)  satisfy  relation  (I)  of  Theorem  I,  §  51  provided,  however, 
that  t  has  only  those  values  for  which  — a-\-^^t  =  l  —  a;  ^  >  0.  ]\Iore- 
over,  when  a  =  1,  relation  (1)6  of  Theorem  VI,  §  55  is  satisfied  for  the  same 
values  of  t  and  in  this  relation  we  have  in  the  present  instance  G2  =  0  or  G2  =  1 
according  as  we  are  dealing  with  (114)  or  (115). 

Again,  if  ^  =  0  in  (115)  the  limit  approached  by  this  expression  as  k  =  <» 
(a'  <a<  h')  will  be  I  +  (a^-'+s  _  ^2^+2)  or  -  ^  +  {0""+'-  -  (3'-"+-)  according  as 
^  >  0  or  /  <  0,  it  being  understood  as  before  that  —  q;+s=<  =  1  —  «• 

Likewise,  if  a  =  1,  other  conditions  remaining  the  same,  the  limit  approached 


156  SUMMABILITY   OF   FOURIER   SeRIES   AND   AlLIED    DEVELOPMENTS 

by  (115)  as  A:  =  00  will  be  -  1  +  {or'^^  -  fi-''+^).  In  both  the  cases  which  thus 
arise  when  ^  =  0  we  evidently  meet  with  an  application  of  the  third  general 
remark  of  §  56  and  we  shall  make  this  application  presently. 

Turning  to  the  other  relations  of  Theorems  I  and  VI  of  §§  51  and  55,  we  see 
that  in  the  present  developments  the  function  (p(n,  a,  t)  is  equal  to 

nQQ^  1      ^"'+'      r    P'(az)Pm-P'mP(az) 

in  the  case  of  (114)  while  for  (115)  the  same  function  reduces  to 

or 

1      /S^-'+i      r   P'((xz)P(l3z)  -  P{^z)P'{az)  , 
(135)      _  (2.  +  2)  ^^-  +  -  ^-,  l^^  p-(.)^  d^' 

according  as  h  =|=  0  or  h  =  0. 

Now,  for  values  of  a,  jS  and  t  in  (126)  we  may  transform  (133),  (134)  and 
(135)  by  use  of  expressions  (122),  (123),  (124)  and  (125)  and  thus  we  find  that, 
exception  being  made  of  the  term  —  {2v  +  2)^-"^^  in  (135),  these  expressions 
all  reduce  to  the  sum  of  the  derivatives  with  respect  to  t  of  the  expression  (132). 
From  this  it  follows  directly  upon  using  the  lemmas  of  the  Appendix  that  the 
above  expressions  satisfy  relations  II  and  III  of  Theorem  I,  §  51 ;  also  that 
when  a  =  1  conditions  (II)  6  and  (111)6  of  Theorem  VI  of  §  55  are  satisfied, 
it  being  understood  throughout  as  before  that  we  are  dealing  only  with  values 
of  t  such  that  —  a+^^f^l  —  o;;^>0. 

Moreover,  if  we  affect  each  of  the  terms  of  (132)  by  the  operation 

1     n 

-E, 

understanding  that  absolute  values  are  taken  under  the  various  integral  signs, 
it  appears  as  in  the  study  of  (93)  that  when  — a+^^i^l  —  a  (^>0) 
relation  (II)'  of  §  52  is  satisfied,  as  also  (11)6'  {h  —  1)  of  §  55. 

It  remains,  then,  merely  to  consider  the  integrals  (114)  and  (115)  when  t 
takes  values  such  that  —  a^t -^  —  a-\-  i,  (^>0)  and  for  this  it  becomes 
necessary,  as  already  noted,  to  use  some  other  expressions  for  P{^z)  and  P'ifiz) 
than  (124)  and  (125),  since  /3  now  takes  values  indefinitely  near  to  zero. 

Considering,  then,  that  t  =  —  a  is  one  of  the  exceptional  points  of  the  type 
mentioned  in  remark  (1)  of  §  56  it  will  now  suffice  for  the  application  of  Theorems 
I,  II,  VI  and  VIII  of  §§  51-55  that  such  additional  conditions  be  placed  upon/(.T) 
that  when  either  of  the  expressions  (133),  (134)  or  (135)  is  multiplied  by /(a  +  t) 
the  absolute  value  of  the  product,  when  considered  for  values  of  t  such  that 


Developments  in  Bessel  Functions  157 

—  a  ^  t  ^  —  a-\-  ^  and  for  all  values  of  n,  may  be  made  uniformly  small  with  ^, 
this  being  true  when  a'  <  a  <  b'  and  when  a  =  1. 

Let  us  now  divide  Cn  into  two  portions  C„"  and  Cn"  the  first  of  these  com- 
prising that  portion  of  the  line  z  =  k  -{-  iy  for  which  \y\<  t],  where  t)  is  an 
arbitrarily  small  positive  quantity  and  the  second  comprising  all  other  portions 

of  Cn'. 

As  regards  the  expressions  (133),  (134)  and  (135)  when  the  integration  is 
performed  over  Cn",  we  have  but  to  make  use  of  the  well  known  formula 

1  C 

PJz)  = T^ I     sin-"  (p  cos  (z  cos  (p)d(p;         v  >  —  h 

to  see  that  when  |jS|<  ^  the  same  expressions  (C„'  now  replaced  by  Cn")  are 
each  of  the  form  (3^'"^^G{a,  /3,  n,  ^,  77)  where  G{a,  /3,  n,  ^,  i))  is  less  in  absolute 
value  than  a  constant  independent  of  a,  (3  and  n. 

In  order  to  study  the  same  expressions  when  the  integration  is  performed  over 
Cn"  we  first  make  the  following  observations: 

Let  us  write  (120)  in  the  form 


/         2.+  1     \ 

\7r  2"+(i/2) 


(136)  P,{z)  =  ^J-  .+(1/2) '■^(^'  ^)^ 

so  that 

/         2?/  +  1     \ 
A{z,  v)  =   {1  +  e(s,  v)]  +  r(2,  v)  tan  Is ^ — tt  I. 

For  all  values  of  z  (real  part  >  0)  lying  upon  Cn"  and  of  modulus  greater 
than  some  fixed  value  zo  >  0  we  see  that  A{z,  v)  remains  less  in  absolute  value 
than  a  constant  Mx.  Moreover,  li  v  {v  ^  neg.  integer)  has  any  value  except 
one  of  the  form  |(1  d=  4w);  n  =  Q,  1,  2,  3,  •  •  •,  the  same  expression  when  con- 
sidered for  values  of  z  (real  or  complex)  as  near  to  zero  as  we  please  remains 
less  in  absolute  value  than  a  constant  il/2  provided  v  ^\.  In  fact,  it  appears 
from  (136)  that  as  z  =  0,  A{z,  v)  will  tend  to  zero  like  2"+^^^-^  since  from  (104)  we 
have 

lim  P^z)  =  o.-p/..   I    IN  ;         p  >  -  I. 


z=0 


2T(j/  +  1)  ' 


Whence,  if  p  has  any  value  =  —  2  except  one  of  the  form  |(4?i  -\-  1);  n  =  0, 
1,  2,  •  •  •,  we  may  write  for  all  values  of  z  upon  Cn"' 

(        21/ +1     \ 

cos  Is T TV   J 

PM  = ^H^iT^) B{z,  p), 

where  B(z,  p)  remains  less  in  absolute  value  than  a  constant  (independent  of  z). 


158  SUMM ABILITY   OF   FoURIER  SeRIES   AND   ALLIED   DEVELOPMENTS 

Similarly,  if  v  has  any  value  >  —  |  except  one  of  the  form  §(4w  —  1) ;  n  =  0, 
1,  2,  •  •  •,  we  may  write  for  all  values  of  ^  upon  C„"' 


sin 


/         2.+  1     \ 


where  B{z,  v)  has  the  properties  just  mentioned. 

It  follows  that  for  all  values  of  z  (real  part  >  0)  upon  C,/"  and  for  all  values 
however  small  of  the  positive  quantity  j8  we  may  write,  provided  v  ^  —  ^ 


cos 


i^^-H^^) 


(137)  P.,(/3^)  = (^^).+(i/o) B(^z,  p) 


or 


sm 


/  2.+  1     \ 

[^z-^^.j 


(138)  P.m  = (^zy+ai2) -^i^^'  ^)' 

where  for  the  indicated  values  of  z  and  /3  the  expression  B(^z,  p)  remains  less 
in  absolute  value  than  a  constant  independent  of  both  j3  and  z  and  where  the 
first  form  can  be  used  in  all  cases  except  when  p  =  ^(4?i  +  1);  ?i  =  0,  1,  2,  •  •  •, 
while  the  second  form  can  be  used  in  all  cases  except  when  p  =  ^(4/1  —  1); 
n=0,  1,  2,  •... 

By  means  of  the  relation 

we  now  obtain,  as  formulae  corresponding  to  (137)  and  (138), 


sin 


(r        2.+  1     \ 


I\'m   =  ^.-(l/2),.+  (l/2)  B{^Z,  P), 

(139) 

(r       2v+1    \ 
cos  \pz J — ''^  I 

Pv  (P2)   =  oi'-(l/2)2"'+(l/2)  -o(p2,  V), 

where  B{^z,  p)  has  the  properties  given  in  connection  with  (137)  and  (138)  and 
where  the  first  or  else  the  second  formula  (and  in  general  both)  can  be  used  for 
any  given  value  of  i'  ^  —  |. 

Now,  if  we  use  in  (133),  (134)  and  (135)  the  forms  (137),  (138)  and  (139) 
(thus  confining  ourselves  to  an  integration  over  Cn")  we  find  as  before  that 
by  taking  j  =  oo  the  complex  integrals  become  simply  those  arising  when,  in- 
stead of  Cn" ,  we  take  as  path  of  integration  the  line  z=  k  -\-  iy,  it  being  under- 
stood that  the  integration  now  consists  of  that  from  y  =  —  ^   to  y  =  —  t] 


Developments  in  Bessel  Functions  159 

together  with  that  from  y  =  rj  to  y  =  go  .  This  statement,  as  in  the  former  case, 
is  seen  to  be  true  either  when  a'  <  a  <  b'  or  a  =  1.  The  resulting  complex 
integrals  thus  take  a  form  analogous  to  (132)  involving  real  integrals  of  the  form 

where  the  expressions  (pi{y),  <P2(y)  and  (psiy)  are  functions  of  k,  a,  j8  and  y  in 
each  of  which  the  numerator  contains,  besides  factors  whose  modulus  is  always 
less  than  a  constant,  terms  in  each  of  which  appears  one  of  the  factors  sinh  ay, 
cosh  ay  while  the  denominator  contains  cosh^  y.  Thus  the  integrals  in  question 
(aside  from  the  factor  |g->'-('/2)  appearing  on  the  outside)  are  always  less  in  absolute 
value  than  a  constant  independent  of  a,  /3  and  k,  it  being  understood  throughout 
that  a'  <  a  <  b'  or  a  =  1  and  1/5 1  <  ^. 

Thus  the  expressions  (104),  (105)  and  (106)  when  considered  for  values  of  jS 
such  that  |i3|  <  ^  are  of  the  form  jS"-^^"^^  H(a,  j(5,  n)  where  H(a,  /5,  n)  is  less  in 
absolute  value  than  a  constant  independent  of  a,  fi  and  n. 

It  follows  therefore  (considering  the  forms  which  we  have  now  obtained  for 
the  expressions  (133),  (134)  and  (135)  when  the  indicated  integration  is  performed 
either  over  Cn"  or  Cn")  that  we  shall  be  able  to  apply  Theorems  I,  II,  VI  and 
VIII  of  §§  51-55  to  the  present  developments  if  we  demand  (in  addition  to 
the  conditions  placed  upon /(a-)  in  the  same  theorems)  that  the  function /(/S)/?""*""'^^ 
be  integrable  in  the  neighborhood  at  the  right  of  the  point  /?  =  0,  it  being  under- 
stood also  that  v  >  —  ^.  In  other  words,  we  need  merely  make  the  additional 
demand  that  x'''^^^''^^f(x)  be  integrable  in  the  neighborhood  at  the  right  of  the 
point  a;  =  0. 

Upon  applying  Theorems  I,  II,  VI  and  VII  of  §§  51-55  and  remarks  (1) 
and  (3)  of  §  56  we  thus  arrive  in  summary  at  the  following  result: 

"  If  f(x)  remains  finite  throughout  the  interval  (0,  1)  with  the  possible  ex- 
ception of  a  finite  number  of  points  and  is  such  that  the  integrals 

(140)        I    .r""^'-''"^  \f(x)  I  dx,  I     |/(a-)  I  dx,         e  arbitrarily  small  and  positive, 

Jo  J  e 

exist  and  if  P^s;)  be  the  function  defined  for  all  values  of  z  and  for  f  >  —  1  by 
the  equation  (104),  then  each  of  the  three  series: 

00 

(2.  +  2)    r  x'^+%v)dx  +  IlqnT^iK'x), 

Jf  n  =  \ 


160  SUMMABILITY   OF   FOUEIER   SERIES   AND   ALLIED   DEVELOPMENTS 

n=l 

in  which  X„,  X„'  and  X„"  represent  respectively  the  nth  positive  roots  of  the 
equations 

P,(2)  =  0,         P/(2)  =  0,        zP/(s)  -  hPM  =  0;         h  =  constant  +  0 
and  in  which 


qn'  =  p^2  J    x'^+'f{x)PX\n'x)dx, 


will  converge  provided  i;  ^  —  ^  at  any  point  x  (0  <  .r  <  1)  in  the  arbitrarily 
small  neighborhood  of  which  f{x)  has  limited  total  fluctuation,  and  the  sum 

will  be 

H/(-^--0)+/(.r  +  0)]. 

IVIoreover,  the  convergence  will  be  uniform  to  the  limit  j{x)  throughout  any 
interval  {a',  h')  enclosed  within  a  second  interval  (ai,  6i)  such  that  0  <  Oi  <  a' 
<  //  <  6i  <  1  provided  /(.r)  is  continuous  throughout  {a',  h')  inclusive  of  the 
end  points  x  =  a',  x  =  h'  and  has  limited  total  fluctuation  throughout  (ai,  hi). 

Also,  if  f{x)  remains  finite  throughout  the  interval  (0,  1)  with  the  possible 
exception  of  a  finite  number  of  points  and  is  such  that  the  integrals  (140)  exist, 
then  each  of  the  three  series  above  {v  ^  —  ^  )will  be  summable  (r  =  1)  at  any 
point  .T  (0  <  ar  <  1)  at  which  the  limits  /(.r  —  0),  j{x  +  0)  exist  and  the  sum 

will  be 

i[/(.i--0)+/Gr  +  0)]. 

Moreover,  the  summability  will  be  uniform  to  the  limit  /(.r)  throughout  any 
interval  (a',  h')  such  that  0  <  a'  <  6'  <  1  provided  that  at  all  points  within 
(a',  h')  inclusive  of  the  end  points  x  =  a',  x  =  b'  the  function /(a-)  is  continuous. 

Under  the  same  conditions  for  f(x)  when  considered  throughout  the  ivhole 
interval  (0,  1)  the  three  series  (i/  ^  —  |),  when  considered  for  the  value  x  =  1, 
will  converge  to  the  respective  limits  0,  /(I  —  0)  and  /(I  —  0)  provided  that 
f(x)  is  of  limited  total  fluctuation  in  the  neighborhood  at  the  left  of  the  point 
x=  1. 

The  same  series  when  considered  for  the  value  .t  =  1  will  be  summable  to  the 
respective  limits  0,  /(I  —  0)  and  /(I  —  0)  whenever  /(I  —  0)  exists." 

67.  If  we  now  introduce  Bessel  functions  into  this  result  through  the  relation 


Developments  in  Bessel  Functions  161 

Py(z)  =  z~''J^(z)  and  then  apply  the  theorem  to  the  function  x'^ix)  instead  of 
f(x)  we  obtain  the  following: 

Theorem.     If  f(x)  remams  finite  throughout  the  interval  (0,  1)  with  the  possible 
exception  of  a  finite  numher  of  points  and  is  such  that  the  integrals 

(141)       I    x^\f{x)\dx,  I     \f{x)\dx;        e  =  arbitrarily  small  positive  constant 

Jo  Je 

exist  and  if  J^{z)  be  Bessel' s  function  of  the  first  kind  of  order  v  then  each  of  the 
three  series 

2gn/v(Xn.X-), 

(2^  +  2)    r  x'^+'f{x)dx  +  flqn'J.iK'x), 

00 
Y.qn"JX^n"x), 

in  which  \n,  X^'  a7id  X,/'  represent  respectively  the  nth  positive  roots  of  the  equations 

JM  =  0, 

^  (S-'J.CZ))   =  2j;(2)  -   vJ^iz)   =  0, 

zJJ{z)  —  (h  +  v)J^{z)  =  0,         h  =  constant  =f=  0, 
and  in  which 


2      r^ 

qn  =  J  ,/^   N2    I      xf{x)J^{\nX)dx, 
qn     ^^  ~T   /-v    /\2    I      •V  C*^/"  vvXn  XjuX, 


qn"  = 


2\n"' 


{h{2p-\-h)+Xn"V.0^n') 


—  J^    xfix)J^(Kn"x)d:i 


will  converge  provided  v  >  —  ^  at  any  point  x  {0  <  x  <  1)  in  the  arbitrarily  small 
neighborhood  of  which  f{x)  has  limited  total  fluctuation,  and  the  sum  will  be 

|[/(^-0)+/(.r  +  0)]. 

Moreover,  the  convergence  will  be  uniform  to  the  limit  f(x)  throughout  any  interval 
(a',  6')  enclosed  within  a  second  interval  (ai,  bi)  such  that  0  <  ai  <  a'  <  b'  <  bi  <  I 
provided  f(x)  is  continuous  throughout  {a',  b')  inclusive  of  the  end  points  x  =  a', 
x  =  b'  and  has  limited  total  fluctuation  throughout  (ai,  by). 

Also,  if  f{x)  remains  finite  throughout  the  interval  (0,  1)  loith  the  possible  ex- 
ception of  a  finite  number  of  points  and  is  such  that  the  integrals  (141)  exist,  then 

12 


162  SUMMABILITY   OF   FoUHIEE   SERIES   AND   AlLIED   DEVELOPMENTS 

each  of  the  three  series  above  (v  >  —  ^)  will  be  summable  (r  =  1)  at  any  point  x 
(0  <  .T  <  1)  at  lohich  the  limits  f{x  —  0),f(x  +  0)  exist  and  the  sum  will  be 

H/G-^ -0) +/(.!• +  0)]. 

Moreover,  the  siimmability  will  be  uniform  (§  45)  to  the  limit  f{x)  throughout 
any  interval  (a',  b')  such  that  0  <  a'  <  6'  <  1  'provided  that  at  all  points  ivithin 
(a',  b')  inclusive  of  the  end  points  x  =  a',  x  =  b'  the  function  f{x)  is  continuous. 

Under  the  same  conditions  for  f(x)  when  considered  throughout  the  whole  interval 
{0,  1),  the  three  series,  ichen  considered  for  the  value  x  =  1  will  converge  to  the 
respective  limits  0,  /(I  —  0)  and  /(I  —  0)  provided  that  f(x)  is  of  limited  total 
fluctuation  in  the  neighborhood  at  the  left  of  the  point  x  =  1. 

The  same  series  ichen  considered  for  the  value  a;  =  1  loill  be  summable  (r  =  1) 
to  the  respective  limits  0,  /(I  —  0)  and  /(I  —  0)  wheriever  /(I  —  0)  exists,  it  being 
always  assumed  that  the  integrals  (141)  exist?'^ 

3.   The  Developments  in  Terms  of  Legendre  Functions. 

68.  We  proceed  to  consider  the  well  known  development 


(142) 


»  2n-\-  1   r^ 

fix)  =  zlqnXnix);         qn  =  ''~^—       f{x)Xn{x)dx, 


in  which  Xn{x)  represents  the  polynomial  of  Legendre  (Zonal  Harmonic)  of  order 
n.  In  the  notation  of  §  60  we  here  have  a  development  of  the  form  (54)  in  which 
H{z,  x)  =  A''z(.t),  a  =  —  l,b  =  1  and  in  which  equation  (53)  becomes 


a^|(i-^-^>l^|  +  *  +  i)-^'  =  °- 


Moreover,  since  z  is  to  take  only  integral  values,  the  equation  u{z)  =  0  must 

22  It  may  be  noted  that  oiu*  results,  in  so  far  as  they  concern  convergence  at  a  special  point 
a;  (0  <  X  <  1),  are  not  in  entire  accord  with  those  of  Dini  ("Serie  di  Fourier,"  pp.  2G6-269). 
In  fact,  instead  of  the  existence  of  the  first  of  the  integrals  (141)  Dini  requires  that  |/(a;)|.r''+a~P, 
where  p  is  the  greater  of  the  two  numbers  v,  \,  shall  be  integrable  in  the  neighborhood  at  the  right 
of  the  point  x  =  0.  This  discrepancy  is  due  chiefly  to  a  slight  error  occurring  in  formula  (95), 
p.  237  of  DiNi's  work,  the  last  term  of  which  should  contain  under  the  integral  sign  e^'^r^""!"" 
instead  of  e~''T''~i~'»,  as  appears  from  the  analysis  on  p.  237.  If  this  formula  (95)  be  altered 
as  just  indicated  and  resulting  changes  be  made  on  pp.  242,  243,  265-269,  we  are  led  to  the  above 
theorem.  This  same  theorem,  so  far  as  it  concerns  convergence,  is  in  accord  with  the  results 
published  in  recent  years  by  Hobson  {Proc.  London  Math.  Soc,  Vol.  7  (1908),  pp.  359-388), 
while,  as  regards  summabihty,  the  theorem  is  in  accord  with  the  results  of  C.  N.  Moore  {Trans. 
Am.  Math.  Soc,  Vol.  10  (1909),  p.  428). 

It  may  also  be  remarked  at  this  point  that,  except  in  the  study  of  uniform  convergence, 
the  results  published  of  late  years  by  Hobson  and  others  respecting  the  convergence  of  Fourier 
series  and  other  developments  in  terms  of  special  normal  functions  were  originally  obtained 
rigorously  for  the  first  time  by  Dini — a  fact  apparently  not  well  imderstood.  See,  however 
NiELSON,  "Handbuch  der  Theorie  der  Cylindcrfunktionen,"  p.  353. 


Developments  in  Legendre  Functions  163 

here  be  regarded  as  given  in  advance  and  may  be  taken  for  example  as 

u(z)  =  sin  TTZ  =  0. 

Furthermore,  we  have  in  the  present  instance  K(x)  =  1  —  x^  so  that  equations 
(60)  become  satisfied  identically  by  taking  h'  =  0,  h  =  0.  However,  since 
K(zL  1)  =  0  it  follows  that  the  general  formulae  of  §  61  for  the  determination 
of  the  integral  (36)  corresponding  to  the  present  development  cannot  be  used. 
It  becomes  necessary,  therefore,  in  order  to  ascertain  whether  this  integral  satis- 
fies the  conditions  of  the  fundamental  Theorem  I,  §51,  to  proceed  independently 
of  such  formulae. 

Now,  the  integral  (36)  here  becomes 

(143)  <p{n,  a,  t)dt  =  ^  E  (2w  +  l)X„(a)        Xn{a  +  t)dt, 

Jo  n=0  Jo 

and  hence  also 

(144)  <p{n,  cx,t)  =  hll  {2n  +  l)Xnia)Xnia  +  t). 

11-0 

We  proceed  to  show  that  the  three  relations  of  the  general  Theorem  I  of  §  51 
are  satisfied  in  the  present  instance,  it  being  understood  that  we  here  have 
a=  -  1,  6  =  1. 

The  values  of  a  and  t  with  which  we  are  concerned  are  such  that 

-  1<  a'  ^  a  ^  h'  <  1, 

We  may  therefore  place  a  =  cos  6' ,  a  -{-  t  =  cos  d  in  which  case  we  have,  as  is 
well  known,^^ 

(145)  X„(cos  ^)X„(cos  6')  =—  j      X„(cos  y)d(p, 

where  cos  y  =  cos  6  cos  6'  +  sin  6  sin  6'  cos  (cp  —  (p'),  it  being  understood  that 
(6,  (p)  and  {6',  tp')  thus  represent  the  polar  spherical  coordinates  of  two  points 
M,  M'  on  the  unit  sphere,  while  y  represents  the  spherical  distance  between  the 
same  points. 

Thus  we  may  write 

I    (pi^n,  a,  t)dl  =  —  J-  E  (2w  +  1)   I    sin  Odd  I      A'„(cos  y)d<p 

Jo  ^TT  ,j=o  J  9'  *'o 

or,  since 

(146)  t  (2n  +  l)X.(cos  7)  =  -  ' •-  -  I  "^r  +  "F^  I ' 
«^  sni  7  [   rty  dy     ] 

"  Cf.  for  example,  Todhunteu's  "Treatise  on  Laplace's  Functions,  Lam6's  Functions,  etc." 
(London,  1875),  §  170. 


164  SUMMABILITY   OF   FOXIRIEE   SeRIES   AND   AlLIED   DEVELOPMENTS 

we  have 

(147)  f  .(.  .,  0.<  =  fj'sin  Mef  j  f  +  ^^  I  ,^. 

Whence,  if  we  denote  by  da  the  element  of  spherical  surface  and  observe  that 
do  is  negative  when  t  is  positive  (i.  e.,  when  6  <  6'),  while  c?0  is  positive  when  t  is 
negative,  we  may  write 

(US)  f.(.«,0.=  .,^//|f  +  ^j^^,, 

in  which  the  upper  or  lower  sign  is  to  be  taken  according  as  t  is  positive  or  nega- 
tive and  where  it  is  understood  that  the  integration  is  extended  over  the  zone 
Ijdng  between  the  parallels  6=6'  and  6=6. 

Let  us  now  choose  a  new  coordinate  system  (7,  \p)  such  that  the  fixed  point 
M'  =  (6',  (p')  becomes  the  point  7  =  0,  while  the  great  circle  through  M'  tangent 
to  the  circle  6=6'  determines  the  points  for  which  x{/  =  0,  Then  da  =  sin  ydydrf/ 
so  that  if  we  represent  by  yi{6')  the  value  of  7  pertaining  to  the  point  upon 
the  circle  6=6'  having  the  (variable)  coordinate  1/'  and  agree  to  place  for  con- 
venience 7„(cos  7)  =  Z„(cos  7)  +  A''„+i(cos  7),  we  may  put  the  equation  (148) 
into  the  form 

J  ^{n,  a,  t)dt  =  T^  J  [F„(cos  7)]^4V#  ±  Ut),'' 

in  which  .9  =  0,  h  =  ir  or  g  =  ir,  h  =  2it  according  as  a  ^  0  or  a  <  0  (0'  ^  7r/2 
or  6'  >  7r/2)  and  in  which  In{t)  is  defined  in  one  of  two  ways  as  follows: 
(a)  li6  <6'  ovd^-K-  6', 

(149)  Ut)  =  l;,£  \ ^n{co^  7)]r=% A 

in  which  c  and  tt  —  c  represent  the  two  values  of  ;/'  determined  by  the  planes  of 
the  two  great  circles  through  the  point  7=0  tangent  to  the  circle  6=6  and  in 
which  72 (^)  and  yz{6)  represent  the  two  values  of  7  pertaining  to  the  points  upon 
the  circle  6=6  having  the  common  coordinate  yp. 
(6)  li6'  <6  K-K-  6', 

(150)  hit)  =  ^f'  Ynicos  yz{6))dyp, 

in  which  yz{6)  represents  the  value  of  7  pertaining  to  the  point  upon  the  circle 
6=6  having  the  coordinate  i/'. 
Upon  writing 

[7„(C0S  7)]^'=i''o^  +  [YnicOS  7)];=o  -   [Fn(C0S  7)]v=v.(«') 

^  We  here  employ  the  common  notation  [f{x)Y'z=a  =  f{b)  —  f{a). 


Developments  in  Legendre  Functions  165 

and  observing  that  F„(cos  0)  =  2,  F„(cos  tt)  =  0,  we  thus  obtain 

(151)  r  ip{:n,  a,  t)dt  =  ±  i  =F^  J     Fn(cos  7i(^'))#  ±  hit). 

In  order  to  show  that  relation  (I)  of  §  51  is  satisfied  it  therefore  suffices  to 
show  that  for  all  values  of  a  and  t  such  that 


(152) 

^    .     ^  .  (6   >  0) 

the  last  two  terms  of  (151)  converge  (n  =  oo)  uniformly  to  zero.     In  doing  this 
we  shall  make  use  of  the  following  two  fundamental  results  respecting  7„(cos  7): 

(A)  For  values  of  7  in  any  interval  such  that  0<^^7^7r—  ^<7r  the 
expression  yn(cos  7)  converges  (n  =  co)  uniformly  to  zero. 

(B)  For  all  values  of  n  we  have  uniformly  hm  F„(cos  7)  =  0 — i.  e.,  corre- 

sponding  to  an  arbitrarily  small  positive  quantity  <j,  one  may  determine  a  second 
positive  quantity  f  independent  of  n  and  such  that  ]  F„(cos  7)  |  <  o-  when 

TT  —    f  =  7  =  TT. 

The  proof  of  {A)  follows  directly  from  the  well-known  fact^^  that  X„(cos  7) 
satisfies  the  indicated  relation,  w^hile  the  proof  of  {B)  may  be  supplied  as  follows : 
From  the  formula^^ 


X., 
we  have 


1    r^  

„(x)  =  -   I     [x  -\-  ^x^  —  1  cos  (pYdip;         —  1  ^  a:  ^  1 

TT  Jo 

Y^{x)  =  -  r    [1  +  .T  +  V.i-'-  -  1  cos  <p\[x  +  V.r'  -  1  cos  ipYd<p. 
Whence, 

\Yn{x)\^-  r  1 1  +  .T  +  V.r-  -  1  COS  <p\d<p^  |1  +  .t|+V1  -  x\ 

TT  Jo 

so  that  for  all  values  of  n  we  have  uniformly 

lim    Yn{x)  =  0        or        Hm  Fn(cos  7)  =  0. 

a:=— 1+0  y=;r— 0 

Results  (A)  and  (B)  being  premised,  we  now  turn  to  the  second  term  appearing 
on  the  right  in  (151)  (which  term,  except  for  the  sign  T,  is  independent  of  t, 
but  depends  upon  a).  Let  us  first  confine  ourselves  to  values  of  a  which  are 
positive  (0  <  a:  <  6') .     For  such  a  value  of  a  the  term  in  question  has  the  form 


1     f 

-   I      F„(cos  yy)d\l/;        71  =  7i(^')- 


25  Cf.  for  example,  Fej£r,  Math.  Annalen,  Vol.  67  (1909),  p.  103. 
2^  Cf.  for  example,  Byerly's  "Fourier  Series,  etc.,"  p.  166. 


166  SUMMABILITY   OF   FoURIEE   SeEIES   .\ND   AlLIED   DEVELOPMENTS 

Omitting  the  factor  =F  l/47r,  let  us  write  this  in  the  form 

F„(cosTi)#+  F„(cos7i)#+  yn(cosTi)# 

+    I  y„(cos  7i)#  +    I        Fn(cos  7l)#, 

where  t;  is  an  arbitrarily  chosen,  small,  positive  quantity. 

Since  |  y„(cos  71)  |  ^  2  whatever  the  values  of  n  and  71,  the  first,  third  and 
last  terms  here  appearing  maj^  be  made  arbitrarily  small  in  absolute  value  with  a 
proper  choice  of  77,  this  being  true  not  only  for  special  values  of  n  and  a,  but  uni- 
formly for  all  values  of  a  such  as  we  are  considering  and  for  all  values  of  n.  After 
7]  has  once  been  chosen,  the  values  of  71  which  enter  into  the  second  and  fourth 
terms  under  the  sign  of  integration  are  seen  (upon  reference  to  the  unit  sphere) 
to  always  be  such  that  0  <  771  ^  71  ^  tt  —  771  <  tt  where  771  depends  only 
upon  77.  From  result  (A)  it  follows  that  the  same  terms  approach  uniformly 
(0  <  a  <  b')  the  limit  zero  as  ?i  =  co . 

Thus  the  second  term  of  (151)  comes  to  have  the  properties  desired. 

We  proceed  to  examine  the  properties  of  the  last  term  in  (151) — i.  e.,  of  the 

expression  In{t)-  Since  t  is  confined  by  relations  (152),  the  angle  d  never  approaches 

(as  t  varies)  nearer  to  6'  (regarded  as  fixed  with  a)  than  some  positive  quantity  k 

which,  if  taken  small  enough,  will  be  independent  of  both  a  and  t.     With  k  thus 

chosen,  it  now  suflSces  to  show  that  for  all  circles  6  such  that  either  0  <  6  <  6'  —  k 

or  6'  -\-  K  <  6  <  T  the  expression  In(t)  converges  (n  =  <x>)  uniformly  to  zero. 

In  showing  this  we  shall  find  it  convenient  to  divide  these  circles  into  three 

classes  as  follows : 

(a)     0  <  d  <  6'  -  K, 

(6)     6'  +  K<  e  <7r-  6', 

(c)     IT  -  d'  <  6  Kir. 

Also,  we  shall  assume  for  the  present  (as  above)  that  B'  <  7r/2  (a  >  0). 

First,  for  the  circles  (a)  we  have  In{i)  defined  by  (149)  in  which  72(^)  and 
yz{d)  are  such  that  k  ^  72  ^  7r/2,  /c  ^  73  ^  20'  —  k  <  tt,  while  c  lies  between 
fixed  limits  dependent  only  upon  e  (as  again  appears  after  noting  the  significance 
of  the  various  letters  upon  the  unit  sphere).  Wlience,  by  result  (A)  we  reach  the 
desired  result  for  the  circles  (a). 

As  regards  the  circles  (6),  let  us  divide  these  into  two  sub-classes  as  follows: 

(by     Tr-e'-v<0<Tr-d', 
(6)"     d'  +  K  <  d  <  T  -  6'  -  V, 
where  77  represents  an  arbitrarily  small  positive  quantity. 


Developments  in  Legendre  Functions  167 

For  the  circles  (6)'  we  have  In(t)  defined  by  (150),  which  may  be  written  in 
the  form 

(154)  i  J""""  F„(cos  y3)drP  +  £  f^  '^  F„(cos  73)^, 

where  co  represents  an  arbitrarily  small  positive  quantity.  Now,  by  choosing  77 
and  o)  each  sufficiently  small,  all  the  values  of  73  entering  into  the  first  term  of 
(154)  may  be  brought  as  near  as  we  please  to  tt,  so  that  in  view  of  result  (B), 
we  conclude  that  the  first  term  of  (154)  may  be  made  arbitrarily  small  in  absolute 
value  by  taking  rj  and  co  sufficiently  small  and  that  this  is  true  uniformly  for  all 
values  of  n.  With  77  and  co  once  fixed,  we  now  observe  that  the  values  of  73 
entering  into  the  second  term  of  (154),  when  considered  for  the  circles  (6)',  never 
approach  nearer  to  tt  than  a  fixed  value  independent  of  6,  while  the  same  values 
of  73  remain  different  by  as  much  as  k  from  0.  Hence,  for  reasons  already  stated 
in  connection  with  the  circles  (a),  the  second  term  of  (154),  when  considered  for 
the  circles  (b)'  approaches  uniformly  the  limit  0. 

With  7)  fixed  as  above,  let  us  now  consider  the  circles  (b)".  Here  again  we 
have  the  form  (150)  for  In(f),  but  the  values  of  73  never  approach  nearer  to  tt 
than  a  certain  positive  value  independent  of  6,  nor  nearer  to  zero  than  k,  so 
that  as  before  we  see  that  uniform  convergence  is  present.  In  summary,  the 
expression  In{t)  has  the  desired  properties  for  all  the  circles  (6). 

W^e  turn  lastly  to  the  circles  (c).  Let  us  divide  these  into  two  sub-classes  as 
follows : 

(c)'     T-  d'  <d^Tr-  6'  -i-  fji, 

(c)"     Tr-e'  +  ix^d<ir, 

where  n  represents  an  arbitrarily  small  positive  constant.  For  the  circles  (c)' 
we  have  In{t)  defined  by  (149)  and  by  taking  /z  sufficiently  small  the  values  of  73 
pertaining  to  these  circles  (c)'  may  be  made  to  differ  by  as  little  as  we  please 
from  TT.  At  the  same  time,  however  small  fx  be  taken,  we  have  72  ^  k  >  0. 
Whence,  using  result  (J5),  we  see  as  before  that  if  v  be  any  preassigned  arbi- 
trarily small  positive  quantity,  we  may  take  ju  so  small  that  for  all  the  circles 
(c)'  we  shall  have  uniformly  |  In(t)  |  <  v.  With  fx  thus  chosen,  let  us  consider  the 
resulting  circles  (c)".  Here  again  we  are  to  use  the  form  (149),  but  the  values 
of  72  and  73  which  enter  lie  between  assignable  limits  m,  n  such  that  m  >  0, 
n  <  TT  (m  =  K,  n  =  IT  —  n).  Hence,  for  the  circles  (c)"  the  expression  7„(0 
has  the  desired  properties,  and  in  summary  we  may  say  that  the  same  is  true  for 
all  the  circles  (c). 

Thus,  relation  (I)  of  §  51  becomes  satisfied  for  all  values  of  a  within  the 
interval  0  ^  a  ^  b'  <  1.  That  it  is  satisfied  also  when  —  l<a'^a;^0 
may  now  be  inferred  as  follows: 


168  SUMMABILITY   OF   FoiIRIER   SeRIES   AND   ALLIED    DEVELOPMENTS 

In  the  present  development  we  have  ip{n,  a,  t)  =  <p(n,  —  a,  —  t)  and  hence 
(155)  I    (p{n,  a,  t)dt  =    I    (p{?i,  —  a,  —  t)dt  =  —    I      (p{n,  —  a,  t)dt. 

Jo  Jo  Jil 

If  a  be  such  that  a'  ^  a  ^  0  it  follows  from  what  we  have  already  shown  that 
the  last  member  of  (155)  will  converge  to  the  limit  ^  or  —  |  according  as  —  1  +  « 
^  —  t^  —  eoT€^  —  t^l-\-a,  and  that  for  all  such  values  of  a  and  t  the 
convergence  will  be  uniform.  This  is,  however,  the  same  as  saying  that  for 
a'  ^  a  ^  0  and  —  1  —  a^t^  —  e  or  e^t^l  —  a  the  first  member  of  (155) 
shall  have  the  properties  desired. 

That  relation  (II)  of  §  51  is  also  satisfied  in  the  present  developments  follows 
from  (151)  together  with  (150).  Thus,  for  all  values  of  a  in  (152)  and  for  all 
values  of  t  such  that  —  €  =  i  =  e  we  have 

I  f\(n.  a,  t)dt  I  g  i  +  1  J'  2#  +  l['  2#  =  2. 

With  regard  to  relation  (III)  of  §  51,  we  note  that  the  function  (p{n,  a,  t) 
of  the  present  development  is  given  by  (144).  Now,  availing  ourselves  of  the 
formula 

9  2^  (2n  +  l)Xn{x')Xn{x)  =  — ^ ''rir:^ 

and  of  the  fact  that  for  large  values  of  ii  the  function  A''n(cos  6)  is  of  the  form^^ 


(— ^y^(n,  0);         \A{n,d)\<  1 


provided  6  lies  in  any  interval  of  the  form  0<  ^  ^  d  ^  ir  —  ^;^  >0,  it  appears 
that  (III)  is  here  satisfied  for  all  values  of  a  and  t  such  that  —  1  <  a'  ^  a  ^  b' 
<1  and  -l-a+^^^^-e,  e^t^l  -  a-  ^  (^>0).  Whether  the 
same  is  also  true  (as  desired  by  (III))  when  t  lies  in  the  intervals  —  1  —  a  ^  t 
^  —  1  —  a  -\-  ^  or  1  —  a—  ^^^^1  —  a  remains  in  doubt,  thus  leading 
eventually  to  an  application  of  remark  (1)  of  §  56.  Due  account  of  this  ex- 
ceptional character  will  be  taken  before  the  final  summary  of  our  results  into  a 
theorem. 

We  turn  to  the  consideration  of  (143)  when  a  =  d=  1.     First,  if  a:  =  1  we 
have 


(156) 


f  cp{n,  1,  t)dt  =  i  E  (2n  +  1)   f  Xn{l  +  t)dt 

Jo  n=0  Jo 


=  -  I  Z)   1    A"„(cos  6)  sin  ddd 

n=0  Jo 


"  Cf.  Fej£r,  I.  c,  p.  103. 


sin  edd 


(158) 


Developments  in  Legendre  Functions  169 

and  we  shall  now  show  that  for  values  of  t  such  that  —  2  +  e  ^  ^  ^  —  e;  i.  e., 
of  6  such  that  OK'n'^B'^ir— r}{r]  arbitrarily  small  but  >  0)  the  last  member 
of  (156)  converges  (uniformly)  to  the  value  —  1  when  n  =  co,  thus  satisfying 
relation  (I) 6  of  §  55  {Gi  =  1)  when  exception  is  there  made  of  the  value  t  =  —  2 
(d  =  tt). 

In  fact,  when  a  =  1  we  have  6'  =  0,  so  that  in  using  (147)  we  have  y  =  6 
while  (p  becomes  independent  of  6.     Thus  we  may  write 

r'  r^  d 

cp{n,  1,  t)dt  =  i        -y^[X„(cos  d)  +  Xn+i(cos  e)]dd 
(157)      •^0  '^y   ^^ 

=   i[^n(C0S  6)  +  X„+i(C0S  e)]l  =    -   1  +  ^rn(C0S  d) . 

The  indicated  statement  thus  follows  upon  noting  the  properties  already  men- 
tioned of  F„(cos  6)  when  0<r]^d^Tr—  r]  (77  >0). 
Again,  if  a  =  —  1  we  may  write 

I    (p{n,  -  1,  t)dt  =  -  ^  Z  (2w  +  1)(-  1)"   I    A^„(cos  6)  sin  ddd 
=  -|Z(2n+l)    f  Z„{cos(7r-0)} 

=  -I  f  F/{cos  (tt-  e)]de 

=    -  UYnicOS    {it  -   e)]]l^^   =    1   -  |7n{cOS    (tT  -   6)], 

from  which  it  appears  that  for  values  of  6  such  that  rj  ^  6  '^  tt  —  r]  (rj  >  0) 
i.  e.,  of  t  such  that  e  ^  t  ^  2  —  e,  the  first  member  of  (158)  converges  uniformly 
(n  =  <x>)  to  the  limit  +  1,  thus  satisfying  relation  (!)„  of  §  55  (6^1  =  1)  when 
exception  is  there  made  of  the  value  t  =  2  (6  =  0). 

Relations  (II)a  and  (11)6  of  §  55  are  evidently  satisfied  as  a  result  of  (157) 
and  (158),  but  relations  (Ill)a  and  (111)6  are  not  satisfied.  For  example,  we 
have 

<pin,  1,  /)  =  ^  E  {2n  +  1)X„(1  +  0  =  I  E  (2«  +  l)X„(cos  d) 

n=0  n=0 

and  as  n  increases  indefinitely  the  right  member  here  appearing  becomes  an 
oscillatory  divergent  series  whatever  the  value  of  d.^^  We  are  here  led,  there- 
fore, not  to  an  application  of  remark  (1)  of  §  5G,  but  rather  to  an  entire  recon- 
sideration of  the  reasoning  by  which  Theorems  (V)  and  (VI)  of  §  55  were  estab- 
lished. In  this  way  we  may  supply  conditions  for  /(.r)  which,  notwithstanding 
the  present  exceptional  character  of  <pin,  1,  t),  will  insure  the  convergence  of  the 
series  (142)  when  x  is  equal  to  either  1  or  —  1. 
28  Cf.  Fej^r,  I.  c,  p.  106. 


170  SUMMABILITY   OF   FoURIER   SERIES   AND   AlLIED   DEVELOPMENTS 

Thus,  if  5„(1)  represents  the  sum  of  the  first  (/i  +  1)  terms  of  the  series  (142) 
when  X  =  1,  we  may  write 

(159)     Snil)  =  J  /(I  +  t)  ^(n,  1,  i)dt  +  j     /(I  +  t)<p{n,  I,  t)dt;         e  >  0. 

Since,  as  already  shown,  relations  (!)&  and  (II)  &  are  satisfied,  it  follows  (of. 
(25))  that  the  first  term  here  appearing  on  the  right  approaches  the  limit /(I  +  0) 
provided  only  that  the  integral 

""   \Kx)\dx 


exists  and  that  f{x)  has  limited  total  fluctuation  in  the  neighborhood  at  the  left 
of  the  point  x  =  1.  It  remains,  therefore,  but  to  impose  such  further  conditions 
upon  /(.r)  that  the  last  term  of  (159)  shall  approach  the  limit  zero  as  n  =  co, 
and  we  shall  now  show  that  this  will  be  the  case  whenever  f{x)  is  of  limited  total 
fluctuation  throughout  the  whole  interval  (—  1,  1). 
First,  let  us  consider  the  integral 


(160) 


r  '  /(I  +  t)cp{n,  1,  t)dL 

«/-2+e 


Considering  that  n  has  any  fixed  value  (positive  integral),  let  us  divide  the 
interval  (—  2  +  e,  —  e)  into  a  certain  number  m  of  parts  such  that  in  each  the 
function  (p{n,  1,  t)  does  not  change  sign.  Let  pi,  jp2,  •'-,  Vm-\  be  the  corre- 
sponding points  of  division.     We  may  then  write 

r'  /(I  +  t)^{n,  1,  t)dt  =  f  f    +   f"  +  •  •  •  +    r  )  /(I  +  t)<p{n,  1,  t)dt 

=  /i  <pdt-\-f2         <pdt+ h/m-i  <pdt+fm  I      cpdt, 

where  (p  =  <p{n,  1,  /)  and  /i,  fo,  fz,  •  •  - ,  fm  are  certain  values  lying  between  the 
upper  and  lower  limits  of /(I  +  0  when  considered  within  the  intervals 

(-2+6,  pi),  (pi,  IH),  •'•,  (Pm-2,  Pm-l),  (Vm-l,   —   i) 

respectively.     Whence,  if  we  let  di,  6-2,  •  •  • ,  6^-1,  6m  be  the  values  of 
(IGl)  I        ip{n,  1,  t)dt 

J_2+€ 

at  the  points  t  =  iH,  t  =  jh,  •  •  •,  t  =  Vm  respectively,  we  may  write 


L 


/(I  +  t)<p{n,  1,  t)dt  =  e,Ui  -  h)  +  Wi  -  /a)  +  •  •  • 

E 

+  ^m— i(/to— 1  ~  Jm)  +  Qmjm' 


Developments  in  Legendre  Functions  171 

Since,  as  already  shown,  the  integral  (157)  when  considered  for  values  of  t 
in  the  interval  —  2  -\-  e^t^  —  e  converges  uniformly  (n  =  oo )  to  the  limit 
—  1,  it  follows  that  the  integral  (160),  when  considered  for  the  same  values  of  t, 
converges  uniformly  to  the  limit  zero.  Whence,  if  a  be  a  preassigned  arbitrarily 
small  positive  quantity  we  shall  have,  at  least  if  n  be  chosen  sufficiently  large, 
I ^1 1  <  0",  I ^2 1  <  0-,  •  •  • ,  \dm\<  cr.  Whence,  also,  if  X  represents  the  upper  limit 
of  /(I  +  t)  between  ^  =  -  2  +  e  and  ^  =  -  €,  and  if  £>«  represents  the  fluctu- 
ation of  /(I  +  t)  in  the  interval  ih  <  t  <  p^+i,  the  last  equation  enables  us  to 
write 

I  r~^  /       "*"'     \ 

/(I  +  t)^{7l,  1,  t)dt     <  C7      X  +  E  £>.       , 
I  «^-2+e  \  s=l  / 

from  which  the  indicated  result  concerning  the  last  term  of  (159)  becomes  evident. 
Similarly,  when  x  =  —  I  we  may  obtain  the  corresponding  result  so  that  the 
discussion  of  the  convergence  of  the  series  (142)  may  now  be  readily  completed, 
both  for  the  case  of  a  point  x  such  that  —  1  <  .r  <  1  and  for  the  end  points 
X  =  ±1,  except  that,  following  remark  (1)  of  §  56,  it  remains  to  consider  the 
integrals 

f{(x^-t)<p{n,a,t)dt,  /        fia+t)<p{n,a,t)dt, 


(162) 


£^     7(1  +  t)cp(n,  1,  t)dt,        J  /(-  1  +  t)cpin,  -  1,  t)dt. 


In  order  to  complete  the  discussion  it  thus  suffices  to  show  that,  at  least  if  ^  be 
taken  sufficiently  small,  each  of  these  integrals  remains  less  in  absolute  value  than 
any  preassigned  positive  quantity  o;  provided  n  be  greater  than  some  fixed 
quantity  N.  Moreover,  in  the  case  of  the  first  two  integrals,  this  property  should 
be  true  uniformly  for  all  values  of  a  such  that  —  I  <  a'  ^a^h'  <  l—i.e., 
the  determination  of  N  should  not  depend  upon  a. 

Taking  the  first  of  the  integrals  (162),  let  us  now  suppose  that  /(.r)  is  of 
limited  total  fluctuation  in  the  neighborhood  at  the  right  of  the  point  a-  =  —  1 
and  hence  that/(a;H-0  has  the  same  property  at  the  right  of  the  point  t=  —  l  —  a. 
Then,  since  we  have  already  shown  that  the  integral  (143)  converges  {n  =  oo) 
to  the  limit  —  ^  and  that  the  convergence  is  uniform  for  all  values  of  a  and  t 
such  that  -l<a'^a^6'<l;  -  \  -  a  ^t  ^  -  e,\t  follows  that  we  may 
treat  the  first  of  the  integrals  (162)  in  the  same  manner  as  we  treated  the 
integral  (160),  thus  showing  that  however  small  the  choice  of  the  positive 
quantity  a,  we  may  determine  a  value  N  (dependent  only  on  a)  such  that 

/(a  +  t)<p{n,  a,  t)dt ,  <  <t     X'  +  Z  D/  )  , 


172  SUMMABILITY   OF   FOUMEK   SeEIES   AND   AlLIED   DEVELOPMENTS 

where  X'  is  the  upper  Hmit  of /(a  +  t)  in  the  interval  —  1  —  a  <  t  <  —  1  —  a-\-  ^ 
and  where 

TO  — 1 

HBs' 

represents  the  sum  of  the  oscillations  of  /(a  +  t)  corresponding  to  a  certain 
division  of  the  same  interval  —  a  sum  which  by  hypothesis  is  less  than  a  constant. 

Similarly,  the  second  of  the  integrals  (162)  is  found  to  have  the  properties 
desired. 

As  regards  the  third  integral,  the  method  just  employed  cannot  here  be  used 
because  we  have  not  investigated  the  convergence  of  the  integral  (157)  when 

—  2^^^  —  2+^.  We  may,  however,  show  as  follows  that  if  f{x)  is  assumed 
to  be  of  limited  total  fluctuation  in  the  neighborhood  at  the  right  of  the  point 
a:  =  —  1  (as  already  implied  in  the  conditions  which  we  have  placed  upon  f{x) 
in  order  that  relation  (Ill)b  be  satisfied)  then  the  third  integral  of  (162)  has 
the  properties  desired.  In  fact,  following  again  remark  (1)  of  §  56,  we  may 
then  show  that  the  integral  in  question  may  be  made  less  in  absolute  value  than 
any  preassigned  positive  quantity  by  taking  ^  sufficiently  small,  this  being  true 
uniformly  for  all  values  of  n  sufficiently  large.  To  see  this,  if  we  let  the  accent 
denote  differentiation  with  respect  to  6,  we  have  from  (157) 

(163)  I         /(I  +  t)ip{n,  1,  t)dt  =  -h  \     /(cos  0)F(cos  e)dd, 

»/— 2  '■'it— I) 

where  77  depends  only  upon  ^  and  vanishes  when  ^  =  0.  Since  f{x)  has  been 
assumed  to  be  of  limited  total  fluctuation  in  the  neighborhood  at  the  right  of  the 
point  X  =  —  \,  the  same  function  is  either  monotone  in  this  interval  or  consists 
of  the  sum  of  a  finite  number  of  such  functions.^^  Evidently,  then,  without  loss 
of  generality  we  may  assume  in  the  study  of  (163)  that  /(cos  6)  is  monotone  in 
the  interval  tt  —  77  <  0  <  tt. 

This  being  the  case,  let  us  apply  the  second  law  of  the  mean  for  integrals  to 
the  second  member  of  (163).     We  obtain 

r ""  /(I + t)<p{n,  1,  t)dt = -  i/(- 1 + ^)  r "'  iv(cos  e)dd 

-  Ui-    1  +  0)     r       Fn'(C0S  d)dd, 

which,  upon  recalling  that  7„(cos  tt)  =  0,  reduces  to 

-  I  (/(-  1  +  ^)  -  /(-  1  +  0) }  F„(cos  (tt  -  77O)  +  i/(-  1  +  ^)  F„(cos  (tt  -  77)), 
and  of  the  two  terms  here  appearing,  the  first,  upon  recalling  that 

I  7n(c0S  (tt  -   771))  I  ^  2, 

23  Cf.  §46,  p.  110. 


Developments  in  Legendre  Functions  173 

may  be  made  arbitrarily  small  in  absolute  value  by  choosing  ^  sufficiently  small, 
while  the  second  (^  having  been  fixed)  vanishes  as  n  =  co,  it  being  observed 
here  that  rj  does  not  depend  upon  n,  so  that  we  are  dealing  with  the  expression 
Ynicos  6),  wherein  6  has  a  fixed  value  such  that  0  <  ^  <  tt. 

Similarly,  it  appears  that  the  last  of  the  integrals  (162)  has  the  properties 
desired  in  case  f(x)  is  assumed  to  be  of  limited  total  fluctuation  in  the  neighbor- 
hood at  the  right  of  the  point  x  =  1. 

In  summary,  then,  we  reach  the  following  theorem  respecting  the  convergence 
of  the  series  (142) : 

Theorem  I.  //  the  function  f(x)  of  the  real  variable  x  satisfies  the  following 
three  conditions: 

(a)  remains  finite  throughout  the  interval  (—  1,  1)  with  the  possible  exception  of 
a  finite  number  of  points ; 

(6)  is  such  that  the  integral 

jy{x)\dx 

exists', 

(c)  is  of  limited  total  fluctuation  in  an  arbitrarily  small  neighborhood  at  the 
right  of  the  point  x  =  —  1  and  in  a  similar  neighborhood  at  the  left  of  the  point 
X  =  \,  then  the  series 


(164) 


<»  2?i  +  1  r^ 

Y^qnXn{x)',  qn  =  Ty I     f{x)Xn{x)dx, 

n=0  ■"  J— I 


in  which  Xn{x)  represents  the  polynomial  of  Legendre  {Zonal  Harmonic)  of  order  n, 
will  converge  at  any  point  x  (—  \  <  x  <  1)  in  the  arbitrarily  small  neighborhood 
of  which  f{x)  has  limited  total  fluctuation,  and  the  sum  will  be 

H/G'^-o)+/(.T  +  o)]. 

Moreover,  the  convergence  will  be  uniform  (§  45)  to  the  limit  f{x)  throughout  any 
interval  (a',  b')  enclosed  within  a  second  interval  (ai,  bi)  such  that  —  1  <  ai  <  a' 
<  b'  <  bi  <  1  provided  that  f(x)  is  continuous  throughout  {a',  b')  inclusive  of  its 
end  points  and  has  limited  total  fluctuation  throughout  (ai,  bi). 

Also,  if  we  replace  conditions  (a),  (b)  and  (c)  by  the  single  more  restrictive  con- 
dition; viz.,  that  f{x)  be  of  limited  total  fluctuation  throughout  the  lohole  interval 
(—  1,  1)  then  the  same  series  will  converge  when  x=  —  lorx=l  and  the  respec- 
tive sums  will  &e/(-  1  +  0),  /(I  -  0).^*^ 

'"  The  results  contained  in  this  theorem,  both  for  the  case  of  an  internal  point  (—  1  <  x  <  1) 
and  for  that  of  the  end  points  x  =  =b  1,  appear  to  have  been  first  established  rigorouslj^  by  Hobson 
{Proc.  London  Math.  Sac,  Vol.  G  (1908),  p.  395.  Ihid.,  Vol.  7  (1909),  p.  39).  Dini's  consideration 
of  the  problem  ("Serie  di  Fourier,  etc.,"  pp.  278-282),  although  outlining  all  the  essential  steps 
of  the  required  analysis,  is  but  fragmentary,  especially  that  which  concerns  the  end  points.  In 
Hobson's  second  paper,  just  noted,  less  stringent  conditions  for  f{x)  are  obtained  than  those 
of  the  Theorem  above,  the  same  resulting  from  an  extended  critical  study  of  the  behavior  of 
Xn{x),  (—  1  ^  x  ^  1)  for  large  values  of  n  {I.  c,  pp.  25-30). 


174  SUMMABILITY   OF   FoUEIER   SerIES   AND   ALLIED   DEVELOPMENTS 

69.  We  proceed  to  consider  the  siimmability  (r  =  1)  of  the  series  (142)  and 
in  so  doing  we  shall  make  use  of  the  following  well  known  result  :^^ 
"  If  we  place 

(165)  sniy)  =  fl{2n+  l)Z„(cos  7) 
and 

(166)  sn'iy)  =  ^-^-^  My)  +  s,iy)  +  •  •  •  +  ^^(7)] 
then  for  large  values  of  71  we  have 


(167)     5„'(7)=i 


7 
4  cos  — 


r     '     ^  ro    •       \^         L\  2/  '         4     I 

Vtt  sin  —  (2  sin  7)^  -" 


[{■-!) 


COS      n  +  -    7-  -J-     +  dn'{y) 


_57r'] 


where  lim  5„'(7)  =  0  uniformly  for  all  e  ^  7  ^  tt  —  €  (e  >  0)." 

71=00 

It  will  thus  appear  that  although  the  summability  (r  =  1)  of  (142)  at  an 
internal  point  (—  1  <  x  <  1)  cannot  be  assured  under  conditions  so  slightly 
limitive  as  those  met  with  in  the  corresponding  studies  of  Fourier  series  (§  46) 
or  the  Bessel  expansions  (§  67),  nor  indeed  under  restrictions  upon  f{x)  which 
are  any  less  than  those  stated  in  Theorem  I  for  convergence  (r  =  0)  at  such  a 
point,  yet  at  the  end  points  a;  =  ±  1  the  conditions  for  summability  may  be 
stated  in  a  less  restrictive  form  than  the  corresponding  ones  in  Theorem  I. 

We  begin  by  noting  that,  as  a  result  of  (144)  and  (145),  the  function  $(«,  n,  t) 
corresponding  to  the  present  development  is  such  that 


where 


$(n,  a,  t)  =  ^^-qj^  [<p{n,  a,  t)  +  <pin  -  l,a,t)  +  ■■•  +  (p(0,  a,  f)], 
(p(n,  a,t)=j-\      2  {2n  +  l)Z„(cos  y)d<p 

^TT  Jo        71  =  0 


the  angle  7  being  here  determined,  as  in  §  68,  through  the  following  relations: 
a  =  cos  6',       a  -\-  t  =  cos  6,       cos  7  =  cos  6  cos  6'  +  sin  6  sin  6'  cos  ((p  —  (p'), 
it  being  understood  that  cp'  is  assigned  any  fixed  value  (0  <  ^'  <  27r)  independent 
of  e. 

Thus  we  may  write 

(168)  <l>(n,  «,  0  =  ^  f^  Sn'{y)dcp, 

where  *„'(7)  is  defined  as  in  (166). 

Now,  when  t  is  such  that  —  €  ^  <  ^  e  (as  occurs  in  relation  (II)  of  the  general 
theorem  of  §  52)  the  corresponding  values  of  7  pertaining  to  the  neighborhood 

"  Cf.  Fej6r,  I.  c,  p.  107. 


Developments  in  Legendre  Functions  175 

of  the  (fixed)  point  {6',  (p')  lie  in  an  interval  of  the  form  0  ^  7  ^  77  where  77 
vanishes  with  e.  Thus,  while  formula  (168)  is  general,  holding  for  all  values  of 
a  and  t  with  which  we  are  concerned  in  applying  the  theorem  of  §  52  to  the 
present  development,  we  are  unable  to  determine  whether  relation  (II)'  of  the 
same  theorem  is  here  satisfied  until  more  than  is  given  by  (177)  is  known  of 
the  behavior  of  Sn'i'^)  for  large  values  of  n.  A  critical  study  of  Sn'{y)  for  0^7 
^  €  is  here  needed  and  such  study  has  apparently  not  yet  been  made. 

Again,  it  cannot  be  argued  from  (167)  and  (168)  that  relation  (III)  of  §  51 
is  here  satisfied  by  ^{n,  a,  t)  (cf.  remark  (2),  §  56).  This  relation,  however,  is 
seen  to  be  satisfied  if  we  confine  ourselves  to  the  intervals  —  1  —  a+^^< 
^  —  6,  e^i^l  —  a  —  ^(^>0)  instead  of  —  1  —  a^^^  —  e,  e^i^l  —  a, 
but  this  is  nothing  more  than  can  be  at  once  inferred  from  the  properties 
already  pointed  out  in  §  68  regarding  the  present  function  <p{n,  a,  t).  It  may 
be  noted  that  if  we  could  show  that  Sn{y)  when  considered  for  all  values  of  n 
and  for  values  of  7  within  the  interval  tt  —  e  ^  7  =  tt  remains  less  than  a  con- 
stant (dependent  only  upon  e)  the  function  $(n,  a,  t)  would  come  to  completely 
satisfy  relation  (III)  as  a  result  of  (167)  and  (168).  That  such  is  true  of  Sniy) 
seems  probable. 

The  conclusion  from  these  remarks  respecting  summability  (r  =  1)  at  an 
internal  point  x(—  l<a;<l)is  therefore  purely  negative,  except  naturally 
that  such  summability  will  necessarily  be  present^^  under  the  conditions  for 
convergence  (r  =  0)  as  given  in  the  theorem  of  the  preceding  §.^^ 

Turning  to  a  consideration  of  the  summability  (r  =  1)  of  the  series  (164) 
when  .T  =  —  1  or  .T  =  1,  we  see  upon  reference  to  the  results  obtained  for 
ipin,  —  1,  t)  and  (p{n,  1,  t)  in  §  68  that  relations  (I)a,  (II)a,  (1)6  and  (II)b  of  §  55 
are  satisfied  by  the  present  functions  <J>(n,  —  1,  t)  and  ^{n,  1,  t)  (regarded  as 
functions  of  the  type  (p  there  indicated)  except  that  doubt  exists  in  the  case 
of  (I)a  and  (1)6  when  t  belongs  to  the  respective  intervals  2  —  ^  ^  t  ^  2, 
—  2^/^  —  2  +  ^(^>0).  In  other  words,  nothing  more  can  be  said  of 
^(n,  —  1,  t)  and  ^{n,  1,  t)  than  was  said  of  (p{n,  —  1,  t)  and  (p{n,  1,  t)  in  §  68. 
This,  however,  is  not  the  case  in  dealing  with  relations  (Ill)a  and  (III)^. 

Thus,  in  (III) 6  we  have  to  consider  the  expression 

$(n,  1,  t)  =  ^^qj^  [<p(n,  1,  0  +  <p{n  -  1,.  1,  0  +  •  •  •  +  <p{0,  1,  /)], 
where 

n 

<p{n,  1,  0  =  ^  E  {2?i  +  l).Y„(cos  6)  =  ^^Sn{d). 

We  may  therefore  write 
(169)  ^{n,  1,  0  =  Wi^)> 

=2  Cf.  §  44. 

33  Cf.  Chapman,  Quart.  Journ.  Math.,  Vol.  43  (1911),  p.  51.  For  summability  (;•  =1) 
Chapman  places  no  restrictions  upon/(x)  at  the  extremities  of  the  interval  (—  1  <  x  <  1)  other 
than  those  for  the  whole  interval. 


176  SUMMABILITY   OF   FOURIER  SeEIES    AND   ALLIED   DEVELOPMENTS 

SO  that  upon  introducing  (167)  we  see  that  (III)^  is  here  satisfied  for  all  values 
of  t  in  the  interval  — 2+^^t^  —  e.  For  the  remaining  values  of  t  with 
which  (111)6  is  concerned,  i.  e.,  —  2  ^  t  ^  —  2  +  ^,  doubt  exists. 

Likewise,  relation  (Ill)a  is  seen  to  be  satisfied  by  $(n,  —  1,  t)  except  possibly 
for  values  of  t  in  the  interval  2  —  ^  ^  t  ^  2. 

From  the  general  theorem  of  §  52  together  with  the  remarks  in  §  56  and  the 
investigations  already  made  in  §  68  of  the  last  two  of  the  integrals  (162)  we 
reach  the  following 

Theorem  IL  If  the  function  f{x)  of  the  real  variable  x  satisfies  conditions 
(a),  (b)  and  (c)  of  the  Theorem  I  (§  68)  then  the  series  (164)  ^vhen  considered  for 
the  values  a*  =  =b  1  will  be  summable  (r  =  1)  to  the  respective  limits  /(I  —  0), 

/(-  1  +  0). 

70.  The  difficulties  which  present  themselves  in  the  study  of  the  summability, 
r  =  1,  of  the  series  (164)  disappear  in  large  measure  when  we  consider  the  same 
problem  with  r  =  2.  This  fact  was  first  pointed  out  by  Fejer^^  who  confined 
himself,  however,  to  functions  f(x)  having  somewhat  greater  limitations  than  we 
shall  here  find  necessary  in  view  of  the  general  theorems  of  §  52.  In  what 
follows  we  shall  make  use  without  further  remark  of  the  following  two  preliminary 
results  which  may  be  found  established  on  pages  81-87  of  Fejer's  original  memoir. 

"  Having  defined  Sniy)  and  Sniy)  as  in  (165)  and  (166),  if  we  place^^ 

(170)  sn'\7)  =  :;^W^y)  +  ^i'(t)  +  •  •  •  +  ^/(t)] 

then 

"(1)  Whatever  the  values  of  n  and  7  (0  <  7  <  tt),  Sn"{y)  is  never  negative. 

"(2)  For  values  of  7  such  that  €  ^  7  ^  tt,  €  being  arbitrarily  small  but  >  0, 
the  expression  Sn"{y)  converges  {n  =  co)  uniformly  to  zero." 

These  results  being  premised,  we  shall  now  endeavor  to  apply  the  general 
theorem  of  §  52  to  the  present  development. 

Just  as  we  found  the  formula  (168)  for  the  function  $(w,  a,  t)  arising  in  the 
study  of  the  summability,  r  =  1,  so  it  appears  that  if  we  represent  by  xl/{n,  a,  t) 
the  corresponding  function  which  arises  when  r  =  2,  we  shall  have 

(171)  Hn,a,t)  =^f    Sn"iy)d<p, 

where  Sn"{y)  is  given  by  (170).     Whence,  upon  using  result  (1)  above,  we  see 
that 


I      \^{n,  a,  t)\dt  =  —    I     \p(n,  a,  t)dt. 


Thus,  in  view  of  the  fact  that  the  function  (pin,  a,  t)  (cf.  (144))  and  hence 

M  Cf.  Math.  Annalen,  Vol.  67  (1909),  pp.  76-109. 

'^  Thus,  8/(7)  comes  to  represent  Holder's  second  mean  for  the  series  (164). 


Developments  in  Legendre  Functions  177 

\p{n,  a,  t)  satisfies  relation  (II)  of  the  theorem  of  §  51  (as  shown  in  §  68)  it  follows 
that  the  present  function  yp{n,  a,  t)  satisfies  relation  (II)'  of  §  52. 

Moreover,  if  we  avail  ourselves  of  result  (2)  above,  it  appears  from  (171)  that 
\p{n,  a,  t),  when  regarded  as  one  of  the  functions  of  the  type  (p{n,  a,  t)  of  §  51, 
satisfies  relation  (III)  of  the  same  §. 

It  remains  but  to  note  that  xl/{n,  a,  t),  when  considered  as  one  of  the  functions 
(p{n,  a,  t)  of  §  51  satisfies  relation  (I)  of  that  §,  as  a  result  of  our  analysis  in  §  68 
in  order  to  see  that  the  conditions  for  the  use  of  the  general  theorem  of  §  51  are 
here  all  satisfied. 

As  regards  the  summability  (r  =  2)  of  the  series  (164)  when  cc  =  ±  1,  it  is 
easily  seen  that  \p{n,  1,  t)  and  ^(w,  —  1,  t)  satisfy  respectively  all  the  conditions 
demanded  by  the  general  theorems  VII  and  VIII  of  §  55.  Thus,  upon  referring 
to  (157)  and  recalling  that  7n(cos  B)  =  Zn+i(cos  6)  +  A^n(cos  d),  we  have  but 
to  make  use  of  result  (2)  above  to  see  that  yp{n,  1,  t)  satisfies  relation  (I) 6  (a  =  —  1, 
h=  l,G2=l)  of  Theorem  VI  (§  55).  Relation  (11)^'  of  Theorem  VIII  (§  55) 
is  also  satisfied,  as  follows  from  the  fact  established  in  §  68  that  ^p{n,  1,  t)  satis- 
fies (11)6,  Theorem  VI  (§  55),  while  from  result  (1)  above,  we  may  write 


XI)  y^O 

\xP{n,  l,t)\dt=   -    1     xP{n,  \,t)dt. 


Finally,  it  follows  from  (169)  that  ^{n,  1,  /)  =  lsn"{d)  so  that  by  applying  result 
(2)  above  we  see  that  yp{n,  1,  t)  satisfies  relation  (111)6,  Theorem  VI  (§  55). 

Upon  noting  the  corresponding  results  concerning  i/'(n,  —  1,  0  and  applying 
the  Theorems  of  §  55  we  reach  in  summary  the  following 

Theorem  III.  If  f(x)  be  any  function  which,  when  considered  throughout  the 
interval  (—  1, 1),  satisfies  conditions  (A)  and  (B)  of  Theorem  I  (§  51)  then  the  series 
(164)  will  be  summable  (r  =  2)  at  any  point  x  {—  I  <  x  <  1)  for  which  the  limits 
f{x  —  0),  f{x  +  0)  exist  and  the  sum  loill  be 

i[/(.T-0)+/(.f +())]. 

Moreover,  the  summability  ivill  be  uniform  (§  45)  to  the  limit  /(.r)  throughout 
any  interval  (a',  b')  such  that  —  I  <  a'  <  b'  <  1  provided  that  at  all  points  within 
(a',  b')  inclusive  of  the  end  points  the  function  f{x)  is  continuous. 

Under  the  same  conditions  {A),  (B)  the  series  lohen  considered  for  the  values 
X  —  —  \  and  X  =  1  will  be  summable  (r  =  2)  to  the  respective  limits  /(—  1  +  0), 
/(I  —  0)  provided  only  that  these  limits  exist. ^^ 

^^  Interesting  results  have  been  obtained  by  Plancherel  {Rend,  del  Cir.  Mat.  di  Palermo, 
Vol.  33  (1912),  pp.  41-66)  relative  to  the  summability  of  the  Legendre  developments  when, 
instead  of  adopting  the  Holder  definition  of  sum,  one  employs  that  of  De  la  Vall6e  Poussin  (see 
footnote,  p.  77). 


APPEiNDIX 

1.  Proof  of  statement  (I),  §  46.  It  is  desirable  for  the  purpose  to  establish  the  following  two 
lemmas: 

Lemma  I.  If  a  and  b  are  any  two  real  numbers  such  that  either  —  7r  +  e^6<a^  —  e 
or  e^a  <  b  ^  w  —  e,  e  being  an  arbitrarily  small  positive  quantity,  and  if  k  is  a  positive  quantity 
which  may  increase  indefinitely  according  to  any  law  whatever,  then 

(1)  Iim   /    —.    -  at  =  0. 

i=oo*^«   sm  t 

and  the  limit  is  approached  uniformly  for  all  the  indicated  values  of  a  and  b. 

In  order  to  establish  this,  let  us  suppose  first  that  a  and  b  are  positive  and  divide  the  cases 
which  may  then  arise  into  three  sets  as  follows: 

(a)a<b^|;         {b)a<^<b;        (c)|^a<6. 

In  (o)  we  have  merelj'^  to  note  that  as  t  varies  from  a  to  6  the  function  1/sin  t  is  always  posi- 
tive and  continually  decreasing  so  that  we  maj'^  apply  the  second  law  of  the  mean  for  integrals 
and  write 

f'^dt  =  J-   Tsin  ktdt  =  -X  rcosfeg-cosA-n 
•^ «  sm  t  sm  a »' «  sm  a  L  k  J 

where  J  is  a  certain  quantity  lying  between  a  and  b.     Hence,  in  (a)  we  shall  have 


(2)  1/^^^^ 

•^  a  sm  i 


"^  sin  kt  ,.    ^2^2 

k  sin  a"^  k  sin  e  ' 

from  which  the  indicated  result  becomes  evident. 
In  (b)  we  write 

(3)  f'^'dt = r^-'^dt + f'^dt, 

^ «■   sm  <  •^'f       sm  i  •^n-/2smf 

where  the  first  integral  of  the  second  member  falls  in  group  (a),  wliile  the  last  one,  after  making 
the  substitution  t  =  w  —  t',  may  be  written 


/'"•/2  sin  kJTT  -  t) 
J n-b         sin  t 


dt. 


In  this  integral  as  t  varies  from  tt  —  6  to  7r/2  the  function  1/sin  t  is  always  positive  and  continually 
decreasing  so  that  we  may  again  apply  the  second  law  of  the  mean  and  write 

jW2sinfcU-0,,  ^     in       .  _  ^  _!_[  COS  kb- COS  k(.- 01 

•'ir-ft        sm  t  smo»'T-6  sm  6  L  k  J' 

where  ■n-  —  6  <  |  <  7r/2. 
WTience, 

I   C"!^  sin  k{w  -  t)  ,\  ^      2 
|*'t-6        sm  t  I      K  sm  e 

after  which  the  indicated  result  becomes  evident  as  before. 
In  (c)  we  have,  after  making  the  substitution  t  —  w  —  t', 

r'j^dt=r;'^^^xp^dt, 

^a  sm  t  J  n-i)         sm  t 

where  c^tt  —  6<7r  —  a^  7r/2.     Hence,  proceeding  as  in  case  (a)  we  may  write  (2). 

178 


Appendix  179 

Upon  noting  that  the  absolute  value  of  the  expression  (1)  remains  unchanged  when  —  a  and 
—  h  are  substituted  for  a  and  b  respectively,  the  Lemma  thus  becomes  completely  established. 

Lemma  II.  If  k  be  a  positive  quantity  increasing  according  to  any  law  whatever  and  if  b  be  a 
constant  {independent  of  k)  and  such  that  0<e<6<ir  —  e,  e  being  an  arbitrarily  small  positive 
quantity,  then 

lim  r'^dt=i,^ 

A;=«''o    smt  2' 

and  the  limit  is  approached  uniformly  for  all  the  indicated  values  of  b. 

In  order  to  prove  this  let  us  indicate  by  k'  the  first  odd  number  equal  to  or  greater  than  k 
and  let  us  place  k  =  k'  —  y  so  that  0  ^  7  ^  2. 

We  may  then  write 

C'^dt  =  r^j^dt+f'^dt, 

«^o   sm  t  'JO    smt  'J  e   sm  t 

in  which  the  last  term,  by  reason  of  Lemma  I,  approaches  uniformly  the  limit  zero  as  A;  =  00. 
Also,  we  may  write 

r^  sin  kt       _    r^  sin  jk'  —  y)t  ,    _    P^  sin  k't  cos  yt       _    p  cos  k't  sin  yl  , 
•^0    sin  i        ~  •^o  sin  i  ~  Jo  sin  i  '^  0         sin  i  ' 

and  by  reason  of  the  general  formula /(5)  =  /(O)  +  5/'(05);  0  <  5  <  1,  we  may  place 

cos  yt  =  1  —  yt  sin  yit 
where  0^71^7  <  2  and  hence 

(4)  f'^^^dt  =   r^^^^^dt  -    ril^^il^yAdt  -  p<^os  k't  sin  yt^^ 

^  '  •'0    sin  i  ^'o    sm «  Jo  sm  i  •'o  sm  < 

Of  the  three  integrals  last  appearing  on  the  right,  the  first  may  be  written  in  the  form 

^  '  \Jo  Je      /   sm<  2       Je       smf      ' 

since,  if  w  be  the  integer  such  that  k'  =  2n  +  1,  we  have 

r'r/2sinfc'i  ,,       f'^fi    ,     ^         n  r\A,      -^ 
\       —. — 7-  dt  =  ]        \  \  -\-   %  coa2nt  \dt  =  -^. 
Jo        sm(  Jo       L         n=i  J  2 

Upon  applying  Lenamal  to  the  last  integral  of  (5)  it  thus  appears  that  the  integral  of  (4)  in 
question  approaches  the  limit  7r/2  as  A;  =  w . 

As  regards  the  last  two  integrals  of  (4),  it  is  at  once  evident  that  each  of  these  may  be  made 
arbitrarily  small  in  absolute  value  with  e  and  with  this  the  proof  of  the  Lemma  becomes  complete. 

The  proof  of  (I)  of  §  46  may  now  be  made  as  follows: 

We  may  write 

.    2n  +  1 


/o^(n,Od<  =  -^(X"Vrj 


^     dt 


.    t 

sm- 

'-«/2sin  (2n  +  1)<  ,,       1  C'l^  sin  (2n  +  l)t 


~       V  '^(12  sint  TT  •'0 


sin  t 


dt 


and  when  — 27r  +  e<<<— ewe  have  —  ir  +  e/2  <  </2  <  —  e/2  so  that  the  first  term  here 
appearing  in  the  last  member  approaches  uniformly  the  limit  zero  when  n  =  <»,  as  appears 
from  Lemma  L  The  last  term  of  the  same  member,  however,  approaches  the  limit  —  ^  as  follows 
from  Lemma  IL 

Siuularly,  when  e  <  t  <  2Tr  —  e  the  desired  result  follows  directly  from  Lemmas  I  and  II 
upon  writing 

1  Cf.  Dim,  Serie  di  Fourier,  etc.,  §  19. 


180  Appendix 

.2/1  +  1 


^  1  f'/2  sin  (2n +  !)<,,    ,    1  f'l^  sin  i2n  +  l)t 


r  <  A^<    w  r  ,  r^  N       2    \,    1  f/''sm(2n+i)<      1  r 


sin  t 


dt. 


sin- 


2.  Proof  0/  statement  (II)  0/  §  46.     We  first  establish  the  following  Lemma: 
Lemma  III.     If  k  is  a  positive  quantity  which  may  increase  indefinitely  according  to  any  law 
whatever  we  may  write^for  any  value  however  great  of  k  and  for  any  value  of  t  such  that  —  7r/2  ^  t  ^  7r/2 : 

sinfci 


\Jo 


sin  t 


dt\  <  w. 


In  fact,  considering  that  A;  has  any  particular  value  among  those  which  it  may  take  and 
considering  first  the  cases  in  which  t  is  positive,  we  observe  that  since  the  function  sin  kt  vanishes 
by  changing  sign  at  the  points  ir/k,  2irlk,  Sir/k,  •  •  •,  while  the  integral 

C  sm  kt  ,^ 

(6)  /„  -^— r  dt 
^^                                                                 ^0   sm  t 

is  positive  from  <  =  Oto^  =  tt/A;,  this  integral  has  maximum  values  at  the  points  tt/A:,  3  tt/A;,  5  tt/Zc,  ••• 
and  minimum  values  at  the  points  2ir/k,  4:ir/k,  •  •  •. 

Moreover,  the  greatest  of  these  maximum  values  is 

(7)  Jo     •siET'^'' 

for  we  may  show  as  follows  that  the  difference  between  any  maximum  value  and  the  next  suc- 
ceeding one  is  positive: 
Let 

(2s  +  l)x  ,  (2s  +  3)7r  /  n    1    n    o  ^      (2s  +  3)7r  _.  7r\ 

tri  =  5^ — ^-^     and    (T2  =  ^ j^         \^s  =0,1,2,3,  ■■•     and    ^-^  <  -  j 

be  any  two  successive  points  belonging  to  the  set  tr/k,  Zwjk,  5-n-/k,  •  •  •.     The  difference  between 
the  corresponding  maximum  values  of  the  integral  in  question  is 

»*<^2  sin  t   „ 

—  at  = 


/  p.  _  p\  sjnkt^^^_  p  sin_fc<^^  ^  _  1    r^-r 
\Jo        Jo     )  sin  f  ♦^''1     sm  <  A;    I 

_  1      ^2.^2).  Sinj  ^  1     ^(2.+2).  _^i__  ^^   ^    _  1      /^2.+2).  g.^  ^  r       1 1  1  ^^_ 

k    \  .     t  k   \  .f  +  x  k    \  \     •     ^        .«  +  ir 

f  sm  r  /  sm — -, —  /„    .s  I  sm  r      sm — ; —  I 

t>'(2«+l)7r  k  »/(2.s+l)7r  k  •^  25+1  ,r  L  A;  fc       J 


In  the  last  integral  here  appearing  the  factor  sin  i  is  negative  (or  zero)  for  all  the  values  of  t  be- 
tween (2s  +  \)ir  and  (2s  +  2)ir  and  since,  for  the  same  values  of  t,  we  have 


A-  ^      k     =  2  ' 
the  factor  appearing  in  square  brackets  in  the  last  integral  is  positive  when 

(2s  +  l)7r<<^(2s  -1-2)7r. 

In  like  manner  it  appears  that  the  least  of  the  minimum  values  of  (6)  is 

J*2T/*sin_fc<  , 
0        sin  t 

and  that  this  value  is  positive  together  with  all  values  of  the  integral  (6)  when  0  <  i  ^  27r/A;. 

Thus,  for  all  values  of  t  such  that  0  <  i  ^  7r/2  the  integral  (6)  is  positive  and  in  summary 
we  may  say  that  the  greatest  absolute  value  of  (6)  when  0  ^  i  ^  7r/2  is  given  by  (7).     But 

2  Cf.  DiNi,  1.  c,  §  18. 


Appendix 


181 


Bin  kh 

kti 


J*"'/*sinfc;  ,,       sin  kit  r^l^,,            kh       r.  ^  .     ^  tt 

—. — 7  dt  =  —. — 7-  /       at  =  T  —. — -  ;  0  <  <i  <  7" 

0       sin  t            sin  ti  Jo                  sin  ti  k 

tx 


and  since  for  such  values  of  <i  this  expression  is  <  1  the  Lemma  now  follows  provided  t  ^  0. 
In  order  to  prove  it  also  for  the  cases  in  which  i  <  0  we  need  but  to  note  that 


X 


'— 'sm  kt  ,\  r'sin  kt 

—. — -  dt\  =  \   I    —. — - 
0      sm  t  «^  0  sm  t 


dt 


By  use  of  Lemma  III  the  proof  of  (II),  §  46,  is  immediate  since  we  have 

2n  +  1 

^'r- sin  {2n  +  l)t 


X  ^^"' ') 


dt 


/^'sii 
«y  0 


t 


.     t 
sm- 


dt 


=  -  r'' 


Sin  t 


dt 


<  1. 


A  possible  choice  of  the  constant  A  of  (II)  is  therefore  A  =  1. 

3.  Proof  of  Statement  (II)',  §  47.     We  shall  here  establish  the  following  general  lemma: 

Lemma  IV.     If  k  =  no.  -\-  ^  where  n  takes  only  positive  integral  values  and  a  and  13  are  any 

two  constants  (independent  of  n)  of  which  a  >  0,  then,  corresponding  to  any  e  >  0  such  that  t  <  7r/2, 

e  <  7r/a,  we  shall  have  for  all  values  of  n  sufficiently  large 


(8) 


I   C    \   t  sin  kt 

—  I         2  

n«^-€    n=o  sin  t 


dt  <g, 


where  g  is  a  certain  constant  independent  of  both  n  and  e. 

Since 

sin  {—  kt)  _  sin  kt 

sin  (—  0         sin  t  ' 
it  will  evidently  suffice  to  prove  the  lemma  for  the  expression 

2.  sin  kt 


(9) 

instead  of  (8). 

Now,  we  have 


nJo     n= 


=0  sin  t 


\dt 


sin  kt 


1 


sin  I        sm  t 
so  that  by  application  of  the  well  known  formulae 


sin  nat  cos  ^t  +  cos  nat  sin  ^t\, 


(10) 


2  sin  nx 

n=0 


sin  —  sin  (n  +  1)  2 


(11) 

we  obtain 

(12) 

where 


%  cos  nxl  = 

n=0 


71X       .  /  ,        -  ^      X 

cos  "2- sin  (n  +  1)  2 
sm^ 


-  C  I    2   -°4'  \dt<-  r  Un,  t)di  +  l£  Mn,  t)dt, 


v^i  = 


n=0 

nat 


sin  i 


.    (w  +  1)q!( 
sin-!^ Ti. 


at 


^2    = 


(n  +  l)a/ 
sm 2— 


sin  t 


sin  0t 

.     at 
sin  -r 


182  Appendix 

Now,  for  the  given  value  of  e  we  maj^  take  n  so  large  that 


and  write 

(13) 

Now,  when 

we  may  write 


(n  +  l)a 

'7r/(n+l)a 


<  e 


0    ^  Jo  J7r/(»+l)a 


.    nat 
sin  t 


0  <t< 

.    nat 
sin— T- 


(?i  +  l)a  ^  *  ^  2 


/    .    nat ' 
na  I  2 


t 


nat 
~2~ 


Tia     TT  _  Trna 

^T*  2  ~"^ 


and  in  like  manner 

(n  +  l)at 
(14) 


a/ 


^^(nj_l)a^ 

<(»  +  ^)   (n  +  i)«^    <(^  +  i)i;      o<^<(;rTI)^' 


Thus  it  appears  that  the  first  term  on  the  right  in  (13)  is  less  than  (7r'?i/8).     Again,  the  second 
term  on  the  right  in  (13)  may  be  put  into  the  form 


2   p 

a  "  f/Ci 


+  l)a 


nat   .    {n  -\r  l)at\ 


sm  -^  sm 


\  sm  i  / 


2 


dt 


.     at       t^' 
^^°2/ 


which  is  less  than 
Thus  we  have 


2   r^  flY^^Ir      in  ^  J[l 

aJrrKn+l)a\2)     t^  =  2  ^'^  ^  ^  ^  2ea' 

nJo  ^'^''-¥+2+2^;^' 


from  which  we  see  that  the  first  term  on  the  right  in  (12)  has  the  property  indicated  of  (8).  Like- 
wise, the  same  is  seen  to  be  true  of  the  second  term  on  the  right  in  (12)  with  which  the  proof 
becomes  complete. 

The  proof  of  (II)'  of  §  47  follows  by  considering  the  special  case  in  which  a  =  1,  j3  =  f. 

4.  Lemma  V.     With  k  defined  as  in  Lemma  IV  we  have 


(15) 


1    /»€     I     n  I 

lim  -  J_      2  cos  kl  \dt=0, 

n=oo  f^  *  I  n=0  I 


where  e  is  any  positive  constant  such  that  e  <  1,  e  <  w/a. 

As  in  the  study  of  (8)  it  will  here  suffice  to  prove  the  lemma  for  the  expression  obtained 
from  (15)  by  replacing  the  —  e  of  the  lower  limit  of  integration  by  0. 

Now,  we  have 

cos  kt  =  cos  nat  cos  fit  —  sin  nat  sin  fit, 

so  that  upon  using  formulae  (10)  and  (11)  we  obtain 

-I     \   •%  coskt\dt<-  /„  Hdt, 

where 

.     (n  +  l)at 


H  = 


at 


Appendix  183 

For  the  given  value  of  e  we  may  take  n  so  large  that 

<  e 


in  +  l)a 
and  write 

(16)  rHdt=  r ""'-'"' Hdt+r     Hdt. 

^      '  Jo  Jo  Jjr/(n+l)a 

From  (14)  it  appears  that  the  first  term  here  appearing  on  the  right  is  less  than  ■K^l2a. 
Again,  the  second  term  on  the  right  in  (16)  may  be  put  into  the  form 


sm- 


nat  I       2       \dt 


which  is  less  than 
Thus  we  have 


a«'rr/(n+l)a  2     I      .     at        t   ' 


2p  /^\dt  ^^         (n  +  l)ea 

aJ7r/(«+l)a  \2jt  a       ^  TT 

2   r^  „  ,,  ^  7r2         2   -       (n  +  l)ea 
-  /„  Hdt  < log  ^ , 

from  which  the  truth  of  the  lemma  becomes  evident. 


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University  of    Michigan  Studies 

HUMANISTIC  SERIES 

General  Editors:    FRANCIS   W.   KELSEY  and  HENRY   A.   SANDERS 

Size,  22.7  X  15.2  cm.     8°,     Bound  in  cloth 

Vol.  I.  Roman  Historical  Sources  and  Institutions.  Edited  by 
Professor  Henry  A.  Sanders,  University  of  Michigan.  Pp.  viii  +  402. 
$2.50  net. 

CONTENTS 

1.  The  Myth  about  Tarpeia:   Professor  Henry  A.  Sanders. 

2.  The  Movements  of  the  Chorus  Chanting  the  Carmen  Saeculare: 

Professor  Walter  Dennison,  Swarthmore  College. 

3.  Studies    in    the    Lives    of    Roman    Empresses,    Julia    Mamaea: 

Professor  Mary  Gilmore  Williams,  Mt.  Holyoke  College. 

4.  The     Attitude    of    Dio     Cassius    toward    Epigraphic    Sources: 

Professor  Duane  Reed  Stuart,  Princeton  University. 

5.  The  Lost  Epitome  of  Livy:   Professor  Henry  A.  Sanders. 

6.  The  Principales  of  the  Early  Empire:    Professor  Joseph  H.  Drake, 

University  of  Michigan. 

7.  Centurions    as    Substitute    Commanders    of    Auxiliary    Corps  ; 

Professor  George  H.  Allen,  University  of  Cincinnati. 


Vol.  n.     Word  Formation  in  PROVENgAL.     By  Professor  Edward  L. 
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Part  H.  A  Study  in  Latin  Abstract  Substantives.  By  Professor 
Manson  A.  Stewart,  Yankton  College.     Pp.  113-78.     $0.40. 

Part  HL  The  Use  of  the  Adjective  as  a  Substantive  in  the  De  Rerum 
Natura  of  Lucretius.  By  Dr.  Frederick  T.  Swan.  Pp.  179-214. 
$0.40. 

Part  IV.  Autobiographic  Elements  in  Latin  Inscriptions.  By  Profes- 
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Parts  Sold  Separately  in  Paper  Covers: 
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Part  II.     The   Myth   of   Hercules   at   Rome.     By   Professor  John   G. 

Winter,  University  of  Michigan.     Pp.  171-273.     $0.50  net. 
Part  III.     Roman  Law  Studies  in  Livy.     By  Professor  Alvin  E.  Evans, 

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Part  IV.     Reminiscences  of  Ennius  in  Silius  Italicus.     By  Dr.  Loura 

B.  Woodruff.     Pp.  355-424.     $0.40  net. 


Vol.  V.  Sources  of  the  Synoptic  Gospels.  By  Rev.  Dr.  Carl  S. 
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263.     $1.30  net. 


Size,  28  X  18.5  cm.     4to. 
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tions in  the  text.     Pp.  viii  +  371.     Bound  in  cloth.     $4.00  net. 


Vol.  VII.  Athenian  Lekythoi  with  Outline  Drawing  in  Matt  Color 
on  a  White  Ground,  and  an  Appendix:  Additional  Lekythoi 
with  Outline  Drawing  in  Glaze  Varnish  on  a  W^hite  Ground. 
By  Arthur  Fairbanks.  With  41  plates.  Pp.  x  +  275.  Bound  in 
Cloth.     $3.50  net. 


Vol.  VIII.     The  Old  Testament  Manuscripts  in  the  Freer  Collec- 
tion.    By  Professor  Henry  A.  Sanders,  University  of  Michigan. 
Part  I.    The  Washington  Manuscript  of  Deuteronomy  and  Joshua. 

With  3  folding  plates  of  pages  of  the  Manuscript  in  facsimile.     Pp. 

vi  +  104.     Paper  covers.     $1.00. 
Part  II.     The  Washington  Manuscript  of  the  Psalms.     {In  Press.) 


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tion.    By  Professor  Henry  A.  Sanders,  University  of  Michigan. 

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Version.  By  Professor  Louis  C.  Karpinski,  University  of  Michigan. 
With  4  plates  showing  pages  of  manuscripts  in  facsimile,  and  25  dia- 
grams in  the  text.     Pp.  vii  +  164.     Paper  covers.     $2.00. 

Part  II.  The  Prodromus  of  Nicholas  Steno's  Latin  Dissertation 
on  a  Solid  Body  Enclosed  by  Natural  Process  within  a  Solid. 
Translated  into  English  by  Professor  John  G.  Winter,  University  of 
Michigan,  with  a  Foreword  by  Professor  William  H.  Hobbs.  With 
2  plates  of  facsimiles,  and  diagrams. 

Part  III.  Vesuvius  in  Antiquity.  Passages  of  Ancient  Authors,  with  a 
Translation  and  Elucidations.     By  Francis  W.  Kelsey.     Illustrated. 


Vol.  XII.     Studies  in  East  Christian  and  Roman  Art. 

Part  I.  East  Christian  Paintings  in  the  Freer  Collection.  By 
Professor  Charles  R.  Morey,  Princeton  University.  With  13  plates 
(10  colored)  and  34  illustrations  in  the  text.  Pp.  xii  +  87.  Bound 
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Part  II.  A  Gold  Treasure  of  the  Late  Roman  Period  from  Egypt. 
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Richard   Gottheil,    Columbia   University.     {In   Preparation.) 


SCIENTIFIC   SERIES 

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Vol.  I.     The  Circulation  and  Sleep.     By  Professor  John  F.  Shepard, 
University  of  Michigan.     Pp.  x  +  83,  with  an  Atlas  of  83  plates, 
bound  separately.     Text  and  Atlas,  $2.50  net. 


Vol.  II.  Studies  on  Divergent  Series  and  Summability.  By  Profes- 
sor Walter  B.  Ford,  University  of  Michigan.  Pp.  xi  +  183,  with  10 
pp.  of  bibliography.    $2.50. 


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HUMANISTIC  PAPERS 

Size,  22.7  X  15.2  cm.     8°.     Bound  in  cloth. 

Latin  and  Greek  in  American  Education,  with  Symposia  on  the 
Value  of  Humanistic  Studies.  Edited  by  Francis  W.  Kelsey. 
Pp.  x  +  396.     $1.50- 

CONTENTS 
The  Present  Position  of  Latin  and  Greek,  the  Value  of  Latin  and 
Greek  as   Educational   Instruments,   the   Nature  of   Culture 
Studies. 

Symposia  on  the  Value  of  Humanistic,  particularly  Classical,  Studies 
AS  A  Preparation  for  the  Study  of  Medicine,  Engineering,  Law 
and  Theology. 

A  Symposium  on  the  Value  of  Humanistic,  particularly  Classical, 
Studies  as  a  Training  for  Men  of  Affairs. 

A  Symposium  on  the  Classics  and  the  New  Education. 

A  Symposium  on  the  Doctrine  of  Formal  Discipline  in  the  Light  of 
Contemporary  Psychology. 


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The  following  pages  contain  advertisements  of  a  few  of  the  Macmillan 
books  on  kindred  subjects. 


The  Theory  of  Errors  and  Least  Squares 

By  LeROY  D.  weld 

Professor  of  Mathematics  in  Coe  College 

202  pages.     12mo. 

This  book  has  been  written  not  only  for  the  use  of  college  stu- 
dents who  have  had  a  short  course  in  calculus,  but  also  for  the  use 
of  scientists  with  a  similar  equipment  in  mathematics.  The  author's 
aim  has  been  to  provide  examples  which  are  clear-cut,  real  and  alive, 
taken  as  far  as  possible  from  actual  and  recent  experience  and  from 
many  departments  of  science.  He  has  been  collecting  these  prob- 
lems for  several  years  from  research-workers,  astronomers,  engineers 
and  chemists,  and  has  assembled  them  here  in  a  form  which  will  be 
found  adequate  for  the  use  of  practical  workers.  To  quote  the 
words  of  a  prominent  mathematician  who  has  read  the  book  in  manu- 
script, "It  avoids  the  congestion  of  technical  terms  which  repels  one 
from  most  of  the  books  on  this  subject,  yet  uses  enough  to  develop 
the  subject  properly.  It  is  written  with  unusual  clearness,  and  has 
a  swing  to  it  that  is  really  quite  charming." 

Memorabilia  Mathematical   or  the 
Philomath's  Ouotation-Book 

By  ROBERT  EDOUARD  MORITZ 

Professor  of  Mathematics  in  the  University  of  Washington 
410  pages.     8vo.     $3.00 

A  timely  and  useful  volume  whose  distinct  purpose  is  to  bring 
together  into  a  single  volume  exact  quotations  in  English  or  English 
translations,  with  their  exact  references,  bearing  on  Mathematics. 
Over  one  thousand  passages  have  been  grouped  under  twenty  heads, 
and  cross-indexed  under  nearly  seven  hundred  topics.  This  book 
should  be  in  the  library  of  every  advanced  student,  instructor  and 
department  of  mathematics. 


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Historical  Introduction  to 
Mathematical  Literature 

By  G.  A.  MILLER 
Professor  of  Mathematics  in  the  University  of  Illinois 

302  pages.     12mo.     $1.60 

This  book  grew  out  of  a  series  of  lectures  which  were  intended  to  sup- 
plement the  work  of  the  regular  mathematical  courses  along  general  his- 
torical lines.  It  aims  to  give  a  brief  account  of  the  most  important  modern 
mathematical  activities,  such  as  the  mathematical  societies,  mathematical 
congresses,  and  periodical  publications  and  reviews,  and  to  exhibit  funda- 
mental results  in  elementary  mathematics  in  the  light  of  their  historical 
development.  In  preparing  it  for  publication,  the  author  aimed  to  meet 
the  needs  of  a  text-book  for  synoptic  and  inspirational  courses  which  can 
be  followed  successfully  by  those  who  have  not  had  extensive  mathematical 
training.  As  the  book  is  written  in  a  popular  style  and  deals  with  many 
present-day  mathematical  questions,  such  as  errors  in  the  current  mathe- 
matical literature,  standing  problems,  and  living  mathematicians,  it  is  be- 
lieved that  it  will  also  be  found  suitable  for  mathematics  teachers'  reading 
clubs.  Most  of  the  book  is  about  mathematics  and  mathematicians,  rather 
than  on  mathematics,  and  is  devoted  to  a  brief  discussion  of  some  of  the 
principal  aids  to  the  student  who  is  just  beginning  to  do  independent 
mathematical  work.  It  includes  a  brief  consideration  of  the  greater  Fer- 
mat  Theorem,  for  the  proof  of  which  a  large  prize  Is  now  outstanding. 

Introduction  to  the  Theory  of  Fourier's 
Series  and  Integrals  and  the  Mathematical 
Theory  of  the  Conduction  of  Heat 

By  H.  S.  CARSLAW 
Professor  of  Mathematics  in  the  University  of  Sydney 

434  pages.     8vo.     $4.25 

An  Introduction  to  the  Theory  of 
Infinite  Series 

By  T.  J.  I'A.  BROMWICH,  M.A.,  F.R.S. 
5n  pages.     8vo.     $4.50 


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MAR  1  3  1967 
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